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%I M1480 #305 Mar 01 2024 09:57:59
%S 1,2,5,15,51,188,731,2950,12235,51822,223191,974427,4302645,19181100,
%T 86211885,390248055,1777495635,8140539950,37463689775,173164232965,
%U 803539474345,3741930523740,17481709707825,81912506777200,384847173838501,1812610804416698
%N Binomial transform of Catalan numbers.
%C Partial sums of A002212 (the restricted hexagonal polyominoes with n cells). Number of Schroeder paths (i.e., consisting of steps U=(1,1),D=(1,-1),H=(2,0) and never going below the x-axis) from (0,0) to (2n-2,0), with no peaks at even level. Example: a(3)=5 because among the six Schroeder paths from (0,0) to (4,0) only UUDD has a peak at an even level. - _Emeric Deutsch_, Dec 06 2003
%C Number of binary trees of weight n where leaves have positive integer weights. Non-commutative Non-associative version of partitions of n. - _Michael Somos_, May 23 2005
%C Appears also as the number of Euler trees with total weight n (associated with even switching class of matrices of order 2n). - _David Garber_, Sep 19 2005
%C Number of symmetric hex trees with 2n-1 edges; also number of symmetric hex trees with 2n-2 edges. A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). A hex tree is symmetric if it is identical with its reflection in a bisector through the root. - _Emeric Deutsch_, Dec 19 2006
%C The Hankel transform of [1, 2, 5, 15, 51, 188, ...] is [1, 1, 1, 1, 1, ...], see A000012 ; the Hankel transform of [2, 5, 15, 51, 188, 731, ...] is [2, 5, 13, 34, 89, ...], see A001519. - _Philippe Deléham_, Dec 19 2006
%C a(n) = number of 321-avoiding partitions of [n]. A partition is 321-avoiding if the permutation obtained from its canonical form (entries in each block listed in increasing order and blocks listed in increasing order of their first entries) is 321-avoiding. For example, the only partition of [5] that fails to be 321-avoiding is 15/24/3 because the entries 5,4,3 in the permutation 15243 form a 321 pattern. - _David Callan_, Jul 22 2008
%C The sequence 1,1,2,5,15,51,188,... has Hankel transform A001519. - _Paul Barry_, Jan 13 2009
%C From _Gary W. Adamson_, May 17 2009: (Start)
%C Equals INVERT transform of A033321: (1, 1, 2, 6, 21, 79, 311, ...).
%C Equals INVERTi transform of A002212: (1, 3, 10, 36, 137, ...).
%C Convolved with A026378, (1, 4, 17, 75, 339, ...) = A026376: (1, 6, 30, 144, ...)
%C (End)
%C a(n) is the number of vertices of the composihedron CK(n). The composihedra are a sequence of convex polytopes used to define maps of certain homotopy H-spaces. They are cellular quotients of the multiplihedra and cellular covers of the cubes. - Stefan Forcey (sforcey(AT)gmail.com), Dec 17 2009
%C a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 2 colors and those at a higher level come in 3 colors. Example: a(4)=15 because we have 2^3 = 8 paths of shape UHD, 2 paths of shape HUD, 2 paths of shape UDH, and 3 paths of shape UHD; here U=(1,1), H=(1,0), and D=(1,-1). - _Emeric Deutsch_, May 02 2011
%C REVERT transform of (1, 2, -3, 5, -8, 13, -21, 34, ... ) where the entries are Fibonacci numbers, A000045. Equivalently, coefficients in the series reversion of x(1-x)/(1+x-x^2). This means that the substitution of the gf (1-x-(1-6x+5x^2)^(1/2))/(2(1-x)) for x in x(1-x)/(1+x-x^2) will simplify to x. - _David Callan_, Nov 11 2012
%C The number of plane trees with nodes that have positive integer weights and whose total weight is n. - _Brad R. Jones_, Jun 12 2014
%C From _Tom Copeland_, Nov 02 2014: (Start)
%C Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
%C Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
%C Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
%C BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)] = (x-x^2) / (1 + x - x^2).
%C Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
%C Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
%C (End)
%C Starting with offset 0, a(n) is also the number of Schröder paths of semilength n avoiding UH (an up step directly followed by a long horizontal step). Example: a(2)=5 because among the six possible Schröder paths of semilength 2 only UHD contains UH. - _Valerie Roitner_, Jul 23 2020
%D J. Brunvoll et al., Studies of some chemically relevant polygonal systems: mono-q-polyhexes, ACH Models in Chem., 133 (3) (1996), 277-298, Eq. 15.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007317/b007317.txt">Table of n, a(n) for n=1..200</a>
%H Roland Bacher and David Garber, <a href="http://arxiv.org/abs/math/0205245">Spindle-configurations of skew lines</a>, arXiv:math/0205245 [math.GT], 2002-2005.
%H Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, <a href="https://arxiv.org/abs/2308.09344">On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks</a>, arXiv:2308.09344 [math.CO], 2023.
%H Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Barry/barry444.html">On the Central Antecedents of Integer (and Other) Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.3.
%H Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2.
%H Andrew M. Baxter and Lara K. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/papers/AvoidingPairs.pdf">Ascent sequences avoiding pairs of patterns</a>, 2015.
%H Janusz Brzozowski and Marek Szykula, <a href="http://arxiv.org/abs/1401.0157">Large Aperiodic Semigroups</a>, arXiv preprint arXiv:1401.0157 [cs.FL], 2013-2014.
%H David Callan, <a href="http://arxiv.org/abs/0802.2275">Pattern avoidance in "flattened" partitions </a>, arXiv:0802.2275 [math.CO], 2008.
%H H. Cambazard and N. Catusse, <a href="http://arxiv.org/abs/1512.06649">Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane</a>, arXiv preprint arXiv:1512.06649 [cs.DS], 2015-2017.
%H Giulio Cerbai, <a href="https://arxiv.org/abs/2401.10027">Pattern-avoiding modified ascent sequences</a>, arXiv:2401.10027 [math.CO], 2024. See p. 17.
%H Giulio Cerbai, Anders Claesson, Luca Ferrari, and Einar Steingrímsson, <a href="https://arxiv.org/abs/2006.05692">Sorting with pattern-avoiding stacks: the 132-machine</a>, arXiv:2006.05692 [math.CO], 2020.
%H Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, <a href="https://doi.org/10.37236/4793">Combinatorial proofs of addition formulas</a>, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
%H Harry Crane, <a href="https://ajc.maths.uq.edu.au/pdf/61/ajc_v61_p057.pdf">Left-right arrangements, set partitions, and pattern avoidance</a>, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00009a021">Enumeration and classification of benzenoid systems. 32. Normal perifusenes with two internal vertices</a>, J. Chem. Inform. Comput. Sci., 32 (1992), 532-540.
%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1016/0166-1280(93)87033-A">Graph-theoretical studies on fluoranthenoids and fluorenoids:enumeration of some catacondensed systems</a>, J. Molec. Struct. (Theochem), 285 (1993), 179-185.
%H S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00021a026">Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes</a>, J. Chem. Inform. Comput. Sci., 34 (1994), 1174-1180.
%H Dennis E. Davenport, Louis W. Shapiro, and Leon C. Woodson, <a href="https://doi.org/10.37236/2034">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
%H Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%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 Paul Duncan and Einar Steingrimsson, <a href="http://arxiv.org/abs/1109.3641">Pattern avoidance in ascent sequences</a>, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
%H Francesc Fite, Kiran S. Kedlaya, Victor Rotger, and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.
%H S. Forcey, <a href="http://www.intlpress.com/HHA/v10/n2/a12/">Quotients of the multiplihedron as categorified associahedra</a>,Homotopy, Homology and Applications, vol. 10(2), 227-256, 2008. [From Stefan Forcey (sforcey(AT)gmail.com), Dec 17 2009]
%H Ira M. Gessel and Jang Soo Kim, <a href="http://arxiv.org/abs/1003.5301">A note on 2-distant noncrossing partitions and weighted Motzkin paths</a>, arXiv:1003.5301 [math.CO], 2010.
%H Ira M. Gessel and Jang Soo Kim, <a href="http://dx.doi.org/10.1016/j.disc.2010.07.017">A note on 2-distant noncrossing partitions and weighted Motzkin paths</a>, Discrete Math. 310 (2010), no. 23, 3421--3425. MR2721104 (2011j:05350). See Eq. (1). - _N. J. A. Sloane_, Jul 05 2014
%H Juan B. Gil and Jordan O. Tirrell, <a href="https://arxiv.org/abs/1806.09065">A simple bijection for classical and enhanced k-noncrossing partitions</a>, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
%H Juan B. Gil and Michael D. Weiner, <a href="https://arxiv.org/abs/1812.01682">On pattern-avoiding Fishburn permutations</a>, arXiv:1812.01682 [math.CO], 2018.
%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See pp. 18-19.
%H U. Grude, <a href="http://www.tfh-berlin.de/~grude/PapRekursion.pdf">Java ist eine Sprache: Rekursive Unterprogramme</a>. See page 4.
%H Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.
%H Frank Harary and Ronald C. Read, <a href="http://dx.doi.org/10.1017/S0013091500009135">The enumeration of tree-like polyhexes</a>, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.
%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=124">Encyclopedia of Combinatorial Structures 124</a>
%H Bradley Robert Jones, <a href="http://summit.sfu.ca/item/14554">On tree hook length formulas, Feynman rules and B-series</a>, Master's thesis, Simon Fraser University, 2014.
%H Hana Kim and R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/hextrees.pdf">A refined enumeration of hex trees and related polynomials</a>, Preprint 2015.
%H Jang Soo Kim, <a href="http://dx.doi.org/10.1016/j.disc.2011.02.020">Bijections on two variations of noncrossing partitions</a>, Discrete Math., 311 (2011), 1057-1063.
%H John W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%H Toufik Mansour and Simone Severini, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.068">Enumeration of (k,2)-noncrossing partitions</a>, Discrete Math., 308 (2008), 4570-4577.
%H Toufik Mansour and Mark Shattuck, <a href="http://arxiv.org/abs/1207.3755">Some enumerative results related to ascent sequences</a>, arXiv preprint arXiv:1207.3755 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 22 2012
%H Igor Pak, <a href="http://dx.doi.org/10.1090/S0002-9939-04-07031-5">Partition identities and geometric bijections</a>, Proc. Amer. Math. Soc. 132 (2004), 3457-3462.
%H Lara K. Pudwell, <a href="http://arxiv.org/abs/1408.6823">Ascent sequences and the binomial convolution of Catalan numbers</a>, arXiv:1408.6823 [math.CO], 2014.
%H Lara Pudwell and Andrew Baxter, <a href="http://faculty.valpo.edu/lpudwell/slides/pp2014_pudwell.pdf">Ascent sequences avoiding pairs of patterns</a>, 2014.
%H Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/ascseq.pdf">Pattern-avoiding ascent sequences</a>, Slides from a talk, 2015.
%H Valerie Roitner, <a href="https://arxiv.org/abs/2008.02240">The vectorial kernel method for walks with longer steps</a>, arXiv:2008.02240 [math.CO], 2020.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Makhin Thitsa and W. Steven Gray, <a href="http://dx.doi.org/10.1137/110852760">On the Radius of Convergence of Interconnected Analytic Nonlinear Input-Output Systems</a>, SIAM Journal on Control and Optimization, Vol. 50, No. 5, 2012, pp. 2786-2813. - From _N. J. A. Sloane_, Dec 26 2012
%H S. H. F. Yan, <a href="http://www.m-hikari.com/ijcms-password2009/17-20-2009/yanIJCMS17-20-2009.pdf">Schröder paths and Pattern Avoiding Partitions</a>, Int. J. Contemp. Math. Sciences, Vol. 4, no. 20, pp. 979-986, 2009.
%F (n+2)*a(n+2) = (6n+4)*a(n+1) - 5n*a(n).
%F G.f.: 3/2-(1/2)*sqrt((1-5*x)/(1-x)) [Gessel-Kim]. - _N. J. A. Sloane_, Jul 05 2014
%F G.f. for sequence doubled: (1/(2*x))*(1+x-(1-x)^(-1)*(1-x^2)^(1/2)*(1-5*x^2)^(1/2)).
%F a(n) = hypergeom([1/2, -n], [2], -4), n=0, 1, 2...; Integral representation as n-th moment of a positive function on a finite interval of the positive half-axis: a(n)=int(x^n*sqrt((5-x)/(x-1))/(2*Pi), x=1..5), n=0, 1, 2... This representation is unique. - _Karol A. Penson_, Sep 24 2001
%F a(1)=1, a(n)=1+sum(i=1, n-1, a(i)*a(n-i)). - _Benoit Cloitre_, Mar 16 2004
%F a(n) = Sum_{k=0..n} (-1)^k*3^(n-k)*binomial(n, k)*binomial(k, floor(k/2)) [offset 0]. - _Paul Barry_, Jan 27 2005
%F G.f. A(x) satisfies 0=f(x, A(x)) where f(x, y)=x-(1-x)(y-y^2). - _Michael Somos_, May 23 2005
%F G.f. A(x) satisfies 0=f(x, A(x), A(A(x))) where f(x, y, z)=x(z-z^2)+(x-1)y^2 . - _Michael Somos_, May 23 2005
%F G.f. (for offset 0): (-1+x+(1-6*x+5*x^2)^(1/2))/(2*(-x+x^2)).
%F G.f. =z*c(z/(1-z))/(1-z) = 1/2 - (1/2)sqrt(1-4z/(1-z)), where c(z)=(1-sqrt(1-4z))/(2z) is the Catalan function (follows from Michael Somos' first comment). - _Emeric Deutsch_, Aug 12 2007
%F G.f.: 1/(1-2x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-3x-x^2/(1-.... (continued fraction). - _Paul Barry_, Apr 19 2009
%F a(n) = Sum_{k, 0<=k<=n} A091965(n,k)*(-1)^k. - _Philippe Deléham_, Nov 28 2009
%F E.g.f.: exp(3x)*(I_0(2x)-I_1(2x)), where I_k(x) is a modified Bessel function of the first kind. - _Emanuele Munarini_, Apr 15 2011
%F If we prefix sequence with an additional term a(0)=1, g.f. is (3-3*x-sqrt(1-6*x+5*x^2))/(2*(1-x)). [See Kim, 2011] - _N. J. A. Sloane_, May 13 2011
%F From _Gary W. Adamson_, Jul 21 2011: (Start)
%F a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
%F 2, 1, 0, 0, 0, 0, ...
%F 1, 2, 1, 0, 0, 0, ...
%F 1, 1, 2, 1, 0, 0, ...
%F 1, 1, 1, 2, 1, 0, ...
%F 1, 1, 1, 1, 2, 1, ...
%F 1, 1, 1, 1, 1, 2, ...
%F ... (End)
%F G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 - A(x)^(n+1))/(1 - A(x)); offset=0. - _Paul D. Hanna_, Nov 07 2011
%F G.f.: 1/x - 1/x/Q(0), where Q(k)= 1 + (4*k+1)*x/((1-x)*(k+1) - x*(1-x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1-x)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 14 2013
%F G.f.: (1-x - (1-5*x)*G(0))/(2*x*(1-x)), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 25 2013
%F Asymptotics (for offset 0): a(n) ~ 5^(n+3/2)/(8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 28 2013
%F G.f.: G(0)/(1-x), where G(k) = 1 + (4*k+1)*x/((k+1)*(1-x) - 2*x*(1-x)*(k+1)*(4*k+3)/(2*x*(4*k+3) + (2*k+3)*(1-x)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 29 2014
%F a(n) = JacobiP(n-1,1,-n-1/2,9)/n. - _Peter Luschny_, Sep 23 2014
%F 0 = +a(n)*(+25*a(n+1) -50*a(n+2) +15*a(n+3)) +a(n+1)*(-10*a(n+1) +31*a(n+2) -14*a(n+3)) +a(n+2)*(+2*a(n+2) +a(n+3)) for all n in Z. - _Michael Somos_, Jan 17 2018
%F a(n+1) = (2/Pi) * Integral_{x = -1..1} (m + 4*x^2)^n*sqrt(1 - x^2) dx at m = 1. In general, the integral, qua sequence in n, gives the m-th binomial transform of the Catalan numbers. - _Peter Bala_, Jan 26 2020
%e a(3)=5 since {3, (1+2), (1+(1+1)), (2+1), ((1+1)+1)} are the five weighted binary trees of weight 3.
%e G.f. = x + 2*x^2 + 5*x^3 + 15*x^4 + 51*x^5 + 188*x^6 + 731*x^7 + 2950*x^8 + 12235*x^9 + ... _Michael Somos_, Jan 17 2018
%p G := (1-sqrt(1-4*z/(1-z)))*1/2: Gser := series(G, z = 0, 30): seq(coeff(Gser, z, n), n = 1 .. 26); # _Emeric Deutsch_, Aug 12 2007
%p seq(round(evalf(JacobiP(n-1,1,-n-1/2,9)/n,99)),n=1..25); # _Peter Luschny_, Sep 23 2014
%t Rest@ CoefficientList[ InverseSeries[ Series[(y - y^2)/(1 + y - y^2), {y, 0, 26}], x], x] (* then A(x)=y(x); note that InverseSeries[Series[y-y^2, {y, 0, 24}], x] produces A000108(x) *) (* _Len Smiley_, Apr 10 2000 *)
%t Range[0, 25]! CoefficientList[ Series[ Exp[ 3x] (BesselI[0, 2x] - BesselI[1, 2x]), {x, 0, 25}], x] (* _Robert G. Wilson v_, Apr 15 2011 *)
%t a[n_] := Sum[ Binomial[n, k]*CatalanNumber[k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 07 2012 *)
%t Rest[CoefficientList[Series[3/2 - (1/2) Sqrt[(1 - 5 x)/(1 - x)], {x, 0, 40}], x]] (* _Vincenzo Librandi_, Nov 03 2014 *)
%t Table[Hypergeometric2F1[1/2, -n+1, 2, -4], {n, 1, 30}] (* _Vaclav Kotesovec_, May 12 2022 *)
%o (PARI) {a(n) = my(A); if( n<2, n>0, A=vector(n); for(j=1,n, A[j] = 1 + sum(k=1,j-1, A[k]*A[j-k])); A[n])}; /* _Michael Somos_, May 23 2005 */
%o (PARI) {a(n) = if( n<1, 0, polcoeff( serreverse( (x - x^2) / (1 + x - x^2) + x * O(x^n)), n))}; /* _Michael Somos_, May 23 2005 */
%o (PARI) /* Offset = 0: */ {a(n)=local(A=1+x);for(i=1,n, A=sum(m=0,n, x^m*sum(k=0,m,A^k)+x*O(x^n))); polcoeff(A,n)} \\ _Paul D. Hanna_
%Y See A181768 for another version. - _N. J. A. Sloane_, Nov 12 2010
%Y First column of triangle A104259. Row sums of absolute values of A091699.
%Y Number of vertices of multiplihedron A121988.
%Y m-th binomial transform of the Catalan numbers: A126930 (m = -2), A005043 (m = -1), A000108 (m = 0), A064613 (m = 2), A104455 (m = 3), A104498 (m = 4) and A154623 (m = 5).
%Y Cf. A055879, A033321, A026376, A026378, A059346, A000045, A000957, A057078, A091867, A104597, A249548, A162326.
%K easy,nonn,nice
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_
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