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%I M2942 N1184 #467 Feb 06 2024 12:12:01
%S 1,3,13,63,321,1683,8989,48639,265729,1462563,8097453,45046719,
%T 251595969,1409933619,7923848253,44642381823,252055236609,
%U 1425834724419,8079317057869,45849429914943,260543813797441,1482376214227923,8443414161166173,48141245001931263
%N Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).
%C Number of paths from (0,0) to (n,n) in an n X n grid using only steps north, northeast and east (i.e., steps (1,0), (1,1), and (0,1)).
%C Also the number of ways of aligning two sequences (e.g., of nucleotides or amino acids) of length n, with at most 2*n gaps (-) inserted, so that while unnecessary gappings: - -a a- - are forbidden, both b- and -b are allowed. (If only other of the latter is allowed, then the sequence A000984 gives the number of alignments.) There is an easy bijection from grid walks given by Dickau to such set of alignments (e.g., the straight diagonal corresponds to the perfect alignment with no gaps). - _Antti Karttunen_, Oct 10 2001
%C Also main diagonal of array A008288 defined by m(i,1) = m(1,j) = 1, m(i,j) = m(i-1,j-1) + m(i-1,j) + m(i,j-1). - _Benoit Cloitre_, May 03 2002
%C So, as a special case of _Dmitry Zaitsev_'s Dec 10 2015 comment on A008288, a(n) is the number of points in Z^n that are L1 (Manhattan) distance <= n from any given point. These terms occur in the crystal ball sequences: a(n) here is the n-th term in the sequence for the n-dimensional cubic lattice. See A008288 for a list of crystal ball sequences (rows or columns of A008288). - _Shel Kaphan_, Dec 26 2022
%C a(n) is the number of n-matchings of a comb-like graph with 2*n teeth. Example: a(2) = 13 because the graph consisting of a horizontal path ABCD and the teeth Aa, Bb, Cc, Dd has 13 2-matchings: any of the six possible pairs of teeth and {Aa, BC}, {Aa, CD}, {Bb, CD}, {Cc, AB}, {Dd, AB}, {Dd, BC}, {AB, CD}. - _Emeric Deutsch_, Jul 02 2002
%C Number of ordered trees with 2*n+1 edges, having root of odd degree, nonroot nodes of outdegree at most 2 and branches of odd length. - _Emeric Deutsch_, Aug 02 2002
%C The sum of the first n coefficients of ((1 - x) / (1 - 2*x))^n is a(n-1). - _Michael Somos_, Sep 28 2003
%C Row sums of A063007 and A105870. - _Paul Barry_, Apr 23 2005
%C The Hankel transform (see A001906 for definition) of this sequence is A036442: 1, 4, 32, 512, 16384, ... . - _Philippe Deléham_, Jul 03 2005
%C Also number of paths from (0,0) to (n,0) using only steps U = (1,1), H = (1,0) and D =(1,-1), U can have 2 colors and H can have 3 colors. - _N-E. Fahssi_, Jan 27 2008
%C Equals row sums of triangle A152250 and INVERT transform of A109980: (1, 2, 8, 36, 172, 852, ...). - _Gary W. Adamson_, Nov 30 2008
%C Number of overpartitions in the n X n box (treat a walk of the type in the first comment as an overpartition, by interpreting a NE step as N, E with the part thus created being overlined). - _William J. Keith_, May 19 2017
%C Diagonal of rational functions 1/(1 - x - y - x*y), 1/(1 - x - y*z - x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%C Dimensions of endomorphism algebras End(R^{(n)}) in the Delannoy category attached to the oligomorphic group of order preserving self-bijections of the real line. - _Noah Snyder_, Mar 22 2023
%D Frits Beukers, Arithmetic properties of Picard-Fuchs equations, Séminaire de Théorie des nombres de Paris, 1982-83, Birkhäuser Boston, Inc.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 593.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81.
%D L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 2, 1999; see Example 6.3.8 and Problem 6.49.
%D D. B. West, Combinatorial Mathematics, Cambridge, 2021, p. 28.
%H Chai Wah Wu, <a href="/A001850/b001850.txt">Table of n, a(n) for n = 0..1308</a> (all terms < 10^1000, first 201 terms from T. D. Noe)
%H M. Abrate, S. Barbero, U. Cerruti, and N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barbero/barbero9.html">Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators</a>, J. Int. Seq. 14 (2011), #11.8.1.
%H B. Adamczewski, J. P. Bell, and E. Delaygue, <a href="http://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "à la Lucas"</a>, arXiv:1603.04187 [math.NT], 2016.
%H J.-M. Autebert, A.-M. Décaillot, and S. R. Schwer, <a href="https://web.archive.org/web/20160303190600/http://smf4.emath.fr/Publications/Gazette/2003/95/smf_gazette_95_51-62.pdf">H.-A. Delannoy et les oeuvres posthumes d'Édouard Lucas</a>, Gazette des Mathématiciens - no 95, Jan 2003 (in French).
%H J.-M. Autebert, M. Latapy, and S. R. Schwer, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00351-5">Le treillis des Chemins de Delannoy</a>, Discrete Math., 258 (2002), 225-234.
%H J.-M. Autebert and S. R. Schwer, <a href="http://dx.doi.org/10.1137/S0895480101387406">On generalized Delannoy paths</a>, SIAM J. Discrete Math., 16(2) (2003), 208-223.
%H Cyril Banderier and Sylviane Schwer, <a href="https://arxiv.org/abs/math/0411128">Why Delannoy numbers?</a>, arXiv:math/0411128 [math.CO], 2004.
%H Cyril Banderier and Sylviane Schwer, <a href="https://doi.org/10.1016/j.jspi.2005.02.004">Why Delannoy numbers?</a>, Journal of Statistical Planning and Inference, 135(1) (2005), 40-54.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, 9 (2006), #06.2.4.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry1/barry242.html">On the Central Coefficients of Riordan Matrices</a>, Journal of Integer Sequences, 16 (2013), #13.5.1.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19 (2016), #16.3.5.
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, 15 (2012), #12.4.8.
%H Paul Barry, <a href="https://arxiv.org/abs/2307.00098">Moment sequences, transformations, and Spidernet graphs</a>, arXiv:2307.00098 [math.CO], 2023.
%H Thomas Baruchel and C. Elsner, <a href="http://arxiv.org/abs/1602.06445">On error sums formed by rational approximations with split denominators</a>, arXiv:1602.06445 [math.NT], 2016.
%H H. Bateman, <a href="/A002692/a002692.pdf">Some problems in potential theory</a>, Messenger Math., 52 (1922), 71-78. [Annotated scanned copy]
%H Raymond A. Beauregard and Vladimir A. Dobrushkin, <a href="http://www.jstor.org/stable/10.4169/math.mag.89.5.359">Powers of a Class of Generating Functions</a>, Mathematics Magazine, 89(5) (2016), 359-363.
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., 22 (2019), #19.3.5.
%H A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv:1507.03227 [math-ph], 2015.
%H J. S. Caughman et al., <a href="http://dx.doi.org/10.1016/j.disc.2007.05.017">A note on lattice chains and Delannoy numbers</a>, Discrete Math., 308 (2008), 2623-2628.
%H Jia-Yu Chen and Chen Wang, <a href="https://arxiv.org/abs/2012.04523">Congruences concerning generalized central trinomial coefficients</a>, arXiv:2012.04523 [math.NT], 2020.
%H Johann Cigler, <a href="http://arxiv.org/abs/1109.1449">Some nice Hankel determinants</a>, arXiv:1109.1449 [math.CO], 2011.
%H Johann Cigler and Christian Krattenthaler, <a href="https://arxiv.org/abs/2003.01676">Hankel determinants of linear combinations of moments of orthogonal polynomials</a>, arXiv:2003.01676 [math.CO], 2020.
%H M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>.
%H F. D. Cunden, <a href="http://arxiv.org/abs/1412.2172">Statistical distribution of the Wigner-Smith time-delay matrix for chaotic cavities</a>, arXiv:1412.2172 [cond-mat.mes-hall], 2014.
%H Emeric Deutsch and B. E. Sagan, <a href="http://arxiv.org/abs/math.CO/0407326">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory 117 (2006), 191-215.
%H Ömür Deveci and Anthony G. Shannon, <a href="https://doi.org/10.20948/mathmontis-2021-50-4">Some aspects of Neyman triangles and Delannoy arrays</a>, Mathematica Montisnigri (2021) Vol. L, 36-43.
%H R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/delannoy.html">Delannoy and Motzkin Numbers</a> [Many illustrations].
%H R. M. Dickau, <a href="/A001850/a001850.gif">The 13 paths in a 4 X 4 grid</a>.
%H T. Doslic, <a href="https://doi.org/10.1007/s10440-009-9515-4">Seven lattice paths to log-convexity</a>, Acta Appl. Mathem. 110(3) (2010), 1373-1392.
%H Tomislav Došlic and Darko Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math. 308(11) (2008), 2182-2212. MR2404544 (2009j:05019) - _N. J. A. Sloane_, May 01 2012
%H D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a>, J. Int. Seq. 13 (2010), #10.9.2.
%H Rui Duarte and António Guedes de Oliveira, <a href="https://www.cmup.pt/sites/default/files/2023-08/GF_LP_corrected_0.pdf">Generating functions of lattice paths</a>, Univ. do Porto (Portugal 2023).
%H M. Dziemianczuk, <a href="http://www.emis.de/journals/INTEGERS/papers/n54/n54.Abstract.html">Generalizing Delannoy numbers via counting weighted lattice paths</a>, INTEGERS, 13 (2013), #A54.
%H M. Dziemianczuk, <a href="http://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv:1410.5747 [math.CO], 2014.
%H James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.
%H Luca Ferrari and Emanuele Munarini, <a href="http://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012.
%H Luca Ferrari and Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">Enumeration of edges in some lattices of paths</a>, J. Int. Seq. 17 (2014), #14.1.5.
%H Seth Finkelstein, <a href="/A001850/a001850.pdf">Letter to N. J. A. Sloane, Mar 24 1990</a>, with attachments.
%H S. Garrabrant and I. Pak, <a href="http://arxiv.org/abs/1407.8222">Counting with irrational tiles</a>, arXiv:1407.8222 [math.CO], 2014.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H Taras Goy and Mark Shattuck, <a href="https://doi.org/10.26493/2590-9770.1645.d36">Determinant identities for the Catalan, Motzkin and Schröder numbers</a>, Art Disc. Appl. Math. (2023).
%H Nate Harman, Andrew Snowden, and Noah Snyder, <a href="https://arxiv.org/abs/2211.15392">The Delannoy Category</a>, arxiv:2211.15392 [math.RT], 2023.
%H Tian-Xiao He, <a href="https://arxiv.org/abs/2011.00173">One-pth Riordan Arrays in the Construction of Identities</a>, arXiv:2011.00173 [math.CO], 2020.
%H M. D. Hirschhorn, <a href="http://web.maths.unsw.edu.au/~mikeh/webpapers/paper68.ps">How many ways can a king cross the board?</a>, Austral. Math. Soc. Gaz., 27 (2000), 104-106.
%H Nick Hobson, <a href="/A001850/a001850.py.txt">Python program for this sequence</a>.
%H P.-Y. Huang, S.-C. Liu, and Y.-N. Yeh, <a href="https://doi.org/10.37236/3693">Congruences of Finite Summations of the Coefficients in certain Generating Functions</a>, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Svante Janson, <a href="https://arxiv.org/abs/1804.06071">Patterns in random permutations avoiding some sets of multiple patterns</a>, arXiv:1804.06071 [math.PR], 2018.
%H S. Kaparthi and H. R. Rao, <a href="http://dx.doi.org/10.1016/0166-218X(91)90055-2">Higher dimensional restricted lattice paths with diagonal steps</a>, Discr. Appl. Math., 31 (1991), 279-289.
%H D. E. Knuth and N. J. A. Sloane, <a href="/A047662/a047662.pdf">Correspondence, December 1999</a>.
%H Daniel Krenn and Jeffrey Shallit, <a href="https://arxiv.org/abs/2401.14231">Strongly k-recursive sequences</a>, arXiv:2401.14231 [cs.FL], 2024.
%H D. F. Lawden, <a href="http://www.jstor.org/stable/3608257">On the Solution of Linear Difference Equations</a>, Math. Gaz., 36 (1952), 193-196.
%H Tamás Lengyel, <a href="http://math.colgate.edu/~integers/v86/v86.pdf">On some p-adic properties and supercongruences of Delannoy and Schröder Numbers</a>, Integers (2021) Vol. 21, #A86.
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 4.
%H Huyile Liang, Jeffrey Remmel, and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 18.
%H Max A. Little and Ugur Kayas, <a href="https://arxiv.org/abs/2107.01752">Polymorphic dynamic programming by algebraic shortcut fusion</a>, arXiv:2107.01752 [cs.DS], 2021.
%H Lily L. Liu, <a href="https://doi.org/10.37236/329">Positivity of three-term recurrence sequences</a>, Electronic J. Combinatorics, 17 (2010), #R57.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878-2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H Lili Mu and Sai-nan Zheng, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Zheng/zheng8.html">On the Total Positivity of Delannoy-Like Triangles</a>, Journal of Integer Sequences, 20 (2017), #17.1.6.
%H Leo Moser, <a href="http://www.jstor.org/stable/3611095">Note 2487: King paths on a chessboard</a>, Math. Gaz., 39 (1955), 54 (one page only).
%H Emanuele Munarini, <a href="http://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%H Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, 9 (2006), #06.2.7.
%H P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., 3 (2000), #00.2.1.
%H R. Pemantle and M. C. Wilson, <a href="http://arXiv.org/abs/math.CO/0003192">Asymptotics of multivariate sequences, I: smooth points of the singular variety</a>, arXiv:math/0003192 [math.CO], 2000.
%H C. de Jesús Pita Ruiz Velasco, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Pita/pita5.html">Convolution and Sulanke Numbers</a>, JIS 13 (2010), #10.1.8.
%H F. Qi, X.-T. Shi and B.-N. Guo, <a href="https://doi.org/10.1007/s13226-016-0211-6">Some properties of the Schroder numbers</a>, Indian J. Pure Appl. Math 47 (4) (2016) 717-732.
%H J. L. Ramírez and V. F. Sirvent, <a href="https://doi.org/10.37236/4618">A Generalization of the k-Bonacci Sequence from Riordan Arrays</a>, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38
%H J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>.
%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.
%H E. Rowland and D. Zeilberger, <a href="http://arxiv.org/abs/1311.4776">A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences</a>, arXiv:1311.4776 [math.CO], 2013.
%H S. Samieinia, <a href="http://www.math.su.se/reports/2008/3">The number of continuous curves in digital geometry</a>, Research Reports in Mathematics, Number 3, 2008.
%H S. R. Schwer and J.-M. Auterbert, <a href="https://www.researchgate.net/publication/30449392_Henri-Auguste_Delannoy_une_biographie">Henri-Auguste Delannoy, une biographie</a>, Math. & Sci. hum. / Mathematical Social Sciences, 43e année, no. 174 (2006), 25-67.
%H Seunghyun Seo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Seo/seo2.html">The Catalan Threshold Arrangement</a>, Journal of Integer Sequences, 20 (2017), #17.1.1.
%H T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series006">The Delannoy numbers</a>.
%H M. Shattuck, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from93to101.pdf">On the zeros of some polynomials with combinatorial coefficients</a>, Annales Mathematicae et Informaticae, 42 (2013), 93-101.
%H Zhao Shen, <a href="https://arxiv.org/abs/2112.11135">On the 3-adic valuation of a class of Apery-like numbers</a>, arXiv:2112.11135 [math.NT], 2021.
%H J. B. Slowinski, <a href="http://dx.doi.org/10.1006/mpev.1998.0522">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution, 10(2) (1998), 264-266.
%H J. B. Slowinski, <a href="http://www.neurociencias.org.ve/cont-cursos-laboratorio-de-neurociencias-luz/Slowinski1998%20phylogenetics.pdf">The Number of Multiple Alignments</a>, Molecular Phylogenetics and Evolution, 10(2) (1998), 264-266.
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%H R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, 5 (2002), #03.1.5.
%H R. A. Sulanke et al., <a href="http://www.jstor.org/stable/3647842">Another description of the central Delannoy numbers, Problem 10894</a>, Amer. Math. Monthly, 110(5) (2003), 443-444.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1009.2486">On Delannoy numbers and Schroeder numbers</a>, Journal of Number Theory 131(12) (2011), 2387-2397. doi:<a href="http://dx.doi.org/10.1016/j.jnt.2011.06.005">10.1016/j.jnt.2011.06.005</a>
%H Marius Tarnauceanu, <a href="http://dx.doi.org/10.1016/j.ejc.2007.12.005">The number of fuzzy subgroups of finite cyclic groups and Delannoy numbers</a>, European Journal of Combinatorics 30(1) (2009), 283-287.
%H Bernhard Von Stengel, <a href="https://doi.org/10.1007/PL00009438">New maximal numbers of equilibria in bimatrix games</a>, Discrete & Computational Geometry 21(4) (1999), 557-568. See Eq. (3.7).
%H Yi Wang, Sai-Nan Zheng, and Xi Chen, <a href="https://doi.org/10.1016/j.disc.2019.04.003">Analytic aspects of Delannoy numbers</a>, Discrete Mathematics 342.8 (2019): 2270-2277.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DelannoyNumber.html">Delannoy Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SchmidtsProblem.html">Schmidt's Problem</a>.
%H W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WOAN/hankel2.html">Hankel Matrices and Lattice Paths</a>, J. Integer Sequences, 4 (2001), #01.1.2.
%H E. X. W. Xia and O. X. M. Yao, <a href="https://doi.org/10.37236/3412">A Criterion for the Log-Convexity of Combinatorial Sequences</a>, The Electronic Journal of Combinatorics, 20 (2013), #P3.
%H Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, <a href="https://doi.org/10.1016/j.laa.2023.12.021">The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths</a>, Linear Alg. Appl. (2024).
%F a(n) = P_n(3), where P_n is n-th Legendre polynomial.
%F G.f.: 1 / sqrt(1 - 6*x + x^2).
%F a(n) = a(n-1) + 2*A002002(n) = Sum_{j} A063007(n, j). - _Henry Bottomley_, Jul 02 2001
%F Dominant term in asymptotic expansion is binomial(2*n, n)/2^(1/4)*((sqrt(2) + 1)/2)^(2*n + 1)*(1 + c_1/n + c_2/n^2 + ...). - _Michael David Hirschhorn_
%F a(n) = Sum_{i=0..n} (A000079(i)*A008459(n, i)) = Sum_{i=0..n} (2^i * C(n, i)^2). - Antti Karttunen, Oct 10 2001
%F a(n) = Sum_{k=0..n} C(n+k, n-k)*C(2*k, k). - _Benoit Cloitre_, Feb 13 2003
%F a(n) = Sum_{k=0..n} C(n, k)^2 * 2^k. - _Michael Somos_, Oct 08 2003
%F a(n - 1) = coefficient of x^n in A120588(x)^n if n>=0. - _Michael Somos_, Apr 11 2012
%F G.f. of a(n-1) = 1 / (1 - x / (1 - 2*x / (1 - 2*x / (1 - x / (1 - 2*x / (1 - x / ...)))))). - _Michael Somos_, May 11 2012
%F INVERT transform is A109980. BINOMIAL transform is A080609. BINOMIAL transform of A006139. PSUM transform is A089165. PSUMSIGN transform is A026933. First backward difference is A110170. - _Michael Somos_, May 11 2012
%F E.g.f.: exp(3*x)*BesselI(0, 2*sqrt(2)*x). - _Vladeta Jovovic_, Mar 21 2004
%F a(n) = Sum_{k=0..n} C(2*n-k, n)*C(n, k). - _Paul Barry_, Apr 23 2005
%F a(n) = Sum_{k>=n} binomial(k, n)^2/2^(k+1). - _Vladeta Jovovic_, Aug 25 2006
%F a(n) = a(-1 - n) for all n in Z. - _Michael Somos_, Sep 23 2006
%F D-finite with recurrence: a(-1) = a(0) = 1; n*a(n) = 3*(2*n-1)*a(n-1) - (n-1)*a(n-2). Eq (4) in _T. D. Noe_'s article in JIS 9 (2006) #06.2.7.
%F Define general Delannoy numbers by (i,j > 0): d(i,0) = d(0,j) = 1 =: d(0,0) and d(i,j) = d(i-1,j-1) + d(i-2,j-1) + d(i-1,j). Then a(k) = Sum_{j >= 0} d(k,j)^2 + d(k-1,j)^2 = A026933(n)+A026933(n-1). This is a special case of the following formula for general Delannoy numbers: d(k,j) = Sum_{i >= 0, p=0..n} d(p, i) * d(n-p, j-i) + d(p-1, i) * d(n-p-1, j-i-1). - _Peter E John_, Oct 19 2006
%F Coefficient of x^n in (1 + 3*x + 2*x^2)^n. - _N-E. Fahssi_, Jan 11 2008
%F a(n) = A008288(A046092(n)). - _Philippe Deléham_, Apr 08 2009
%F G.f.: 1/(1 - x - 2*x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - _Paul Barry_, May 28 2009
%F G.f.: d/dx log(1/(1 - x*A001003(x))). - _Vladimir Kruchinin_, Apr 19 2011
%F G.f.: 1/(2*Q(0) + x - 1) where Q(k) = 1 + k*(1-x) - x - x*(k + 1)*(k + 2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013
%F a(n) = Sum_{k=0..n} C(n,k) * C(n+k,k). - _Joerg Arndt_, May 11 2013
%F G.f.: G(0), where G(k) = 1 + x*(6 - x)*(4*k + 1)/(4*k + 2 - 2*x*(6-x)*(2*k + 1)*(4*k + 3)/(x*(6 - x)*(4*k + 3) + 4*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 22 2013
%F G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 16 2013
%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(6 - x)*(2*k + 1)/(x*(6 - x)*(2*k + 1) + 2*(k + 1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 17 2013
%F a(n)^2 = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k) = A243949(n). - _Paul D. Hanna_, Aug 17 2014
%F a(n) = hypergeom([-n, -n], [1], 2). - _Peter Luschny_, Nov 19 2014
%F a(n) = Sum_{k=0..n/2} C(n-k,k) * 3^(n-2*k) * 2^k * C(n,k). - _Vladimir Kruchinin_, Jun 29 2015
%F a(n) = A049600(n, n-1).
%F a(n) = Sum_{0 <= j, k <= n} (-1)^(n+j)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A126086 and A274668. - _Peter Bala_, Jan 15 2020
%F a(n) ~ c * (3 + 2*sqrt(2))^n / sqrt(n), where c = 1/sqrt(4*Pi*(3*sqrt(2)-4)) = 0.572681... (Banderier and Schwer, 2005). - _Amiram Eldar_, Jun 07 2020
%F a(n+1) = 3*a(n) + 2*Sum_{l=1..n} A006318(l)*a(n-l). [Eq. (1.16) in Qi-Shi-Guo (2016)]
%F a(n) ~ (1 + sqrt(2))^(2*n+1) / (2^(5/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 09 2023
%e G.f. = 1 + 3*x + 13*x^2 + 63*x^3 + 321*x^4 + 1683*x^5 + 8989*x^6 + ...
%p seq(add(multinomial(n+k,n-k,k,k),k=0..n),n=0..20); # _Zerinvary Lajos_, Oct 18 2006
%p seq(orthopoly[P](n,3), n=0..100); # _Robert Israel_, Nov 03 2015
%t f[n_] := Sum[ Binomial[n, k] Binomial[n + k, k], {k, 0, n}]; Array[f, 21, 0] (* Or *)
%t a[0] = 1; a[1] = 3; a[n_] := a[n] = (3(2 n - 1)a[n - 1] - (n - 1)a[n - 2])/n; Array[a, 21, 0] (* Or *)
%t CoefficientList[ Series[1/Sqrt[1 - 6x + x^2], {x, 0, 20}], x] (* _Robert G. Wilson v_ *)
%t Table[LegendreP[n, 3], {n, 0, 22}] (* _Jean-François Alcover_, Jul 16 2012, from first formula *)
%t a[n_] := Hypergeometric2F1[-n, n+1, 1, -1]; Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Feb 26 2013 *)
%t a[ n_] := With[ {m = If[n < 0, -1 - n, n]}, SeriesCoefficient[ (1 - 6 x + x^2)^(-1/2), {x, 0, m}]]; (* _Michael Somos_, Jun 10 2015 *)
%o (PARI) {a(n) = if( n<0, n = -1 - n); polcoeff( 1 / sqrt(1 - 6*x + x^2 + x * O(x^n)), n)}; /* _Michael Somos_, Sep 23 2006 */
%o (PARI) {a(n) = if( n<0, n = -1 - n); subst( pollegendre(n), x, 3)}; /* _Michael Somos_, Sep 23 2006 */
%o (PARI) {a(n) = if( n<0, n = -1 - n); n++; subst( Pol(((1 - x) / (1 - 2*x) + O(x^n))^n), x, 1);} /* _Michael Somos_, Sep 23 2006 */
%o (PARI) a(n)=if(n<0, 0, polcoeff((1+3*x+2*x^2)^n, n)) \\ _Paul Barry_, Aug 22 2007
%o (PARI) /* same as in A092566 but use */
%o steps=[[1,0], [0,1], [1,1]]; /* _Joerg Arndt_, Jun 30 2011 */
%o (PARI) a(n)=sum(k=0,n,binomial(n,k)*binomial(n+k,k)); \\ _Joerg Arndt_, May 11 2013
%o (PARI) x='x+O('x^100); Vec(1/sqrt(1 - 6*x + x^2)) \\ _Altug Alkan_, Oct 17 2015
%o (Python) # from Nick Hobson.
%o def f(a, b):
%o if a == 0 or b == 0:
%o return 1
%o return f(a, b - 1) + f(a - 1, b) + f(a - 1, b - 1)
%o [f(n, n) for n in range(7)]
%o (Python)
%o from gmpy2 import divexact
%o A001850 = [1, 3]
%o for n in range(2,10**3):
%o A001850.append(divexact(A001850[-1]*(6*n-3)-(n-1)*A001850[-2],n))
%o # _Chai Wah Wu_, Sep 01 2014
%o (Maxima) a(n):=coeff(expand((1+3*x+2*x^2)^n),x,n);
%o makelist(a(n),n,0,12); /* _Emanuele Munarini_, Mar 02 2011 */
%o (Sage)
%o a = lambda n: hypergeometric([-n, -n], [1], 2)
%o [simplify(a(n)) for n in range(23)] # _Peter Luschny_, Nov 19 2014
%Y Cf. A008288, bisection of A026003, A027618, A047665, A052141, A084773, A152250, A109980, A000129, A078057, A243949.
%Y Main diagonal of A064861.
%Y Column k=2 of A262809 and A263159.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E New name and reference Sep 15 1995
%E Formula and more references from _Don Knuth_, May 15 1996
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