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%I M2995 #226 Dec 12 2023 14:33:10
%S 1,3,15,84,495,3003,18564,116280,735471,4686825,30045015,193536720,
%T 1251677700,8122425444,52860229080,344867425584,2254848913647,
%U 14771069086725,96926348578605,636983969321700,4191844505805495,27619435402363035
%N a(n) = binomial(3n,n).
%C Number of paths in Z X Z starting at (0,0) and ending at (3n,0) using steps in {(1,1),(1,-2)}.
%C Number of even trees with 2n edges and one distinguished vertex. Even trees are rooted plane trees where every vertex (including root) has even degree.
%C Hankel transform is 3^n*A051255(n), where A051255 is the Hankel transform of C(3n,n)/(2n+1). - _Paul Barry_, Jan 21 2007
%C a(n) is the number of stack polyominoes inscribed in an (n+1) X (n+1) box. Equivalently, a(n) is the number of unimodal compositions with n+1 parts in which the maximum value of the parts is n+1. For instance, for n = 2, we have the following compositions: (3,3,3), (2,3,3), (1,3,3), (3,3,1), (3,3,2), (2,2,3), (1,2,3), (2,3,1), (1,1,3), (1,3,1), (3,1,1), (2,3,2), (1,3,2), (3,2,1), (3,2,2). - _Emanuele Munarini_, Apr 07 2011
%C Conjecture: a(n)==3 (mod n^3) iff n is an odd prime. - _Gary Detlefs_, Mar 23 2013. The congruence a(p) = binomial(3*p,p) = 3 (mod p^3) for odd prime p is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - _Peter Bala_, Dec 28 2014
%C In general, C(k*n,n) = C(k*n-1,n-1)*C((k*n)^2,2)/(3*n*C(k*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H G. C. Greubel, <a href="/A005809/b005809.txt">Table of n, a(n) for n = 0..1000</a>[Terms 0 to 100 computed by T. D. Noe; terms 101 to 1000 by G. C. Greubel, Jan 14 2017]
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%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), Article 13.5.1.
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H Naiomi Tuere Cameron, <a href="https://web.archive.org/web/20190331234929/https://www.math.hmc.edu/~cameron/dissertation.pdf">Random walks, trees and extensions of Riordan group techniques</a>, Dissertation, Howard University, 2002.
%H Tom C. Copeland, <a href="http://tcjpn.wordpress.com/2012/06/13/depressed-equations-and-generalized-catalan-numbers/">Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers</a>
%H Maciej 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 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>.
%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>, 6ed, 2013, p. 436.
%H Yaping Liu, <a href="http://www.ripublication.com/gjpam22/gjpamv18n1_05.pdf">On the Recursiveness of Pascal Sequences</a>, Global J. of Pure and Appl. Math. (2022) Vol. 18, No. 1, 71-80.
%H Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/557230/ordinary-generating-function-for-binom3nn/714453#714453">Ordinary generating function for binom(3n,n)</a>, Nov 2013.
%H Romeo Meštrović, <a href="http://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011)</a>, arXiv:1111.3057 [math.NT], 2011.
%H Wojciech Młotkowski and Karol A. Penson, <a href="https://doi.org/10.1142/S0219025714500143">Probability distributions with binomial moments</a>, Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 17, No. 2 (2014), 1450014; <a href="http://arxiv.org/abs/1309.0595">arXiv preprint</a>, arXiv:1309.0595 [math.PR], 2013.
%H Khodabakhsh Hessami Pilehrood and Tatiana Hessami Pilehrood, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Pilehrood/pile5.html">Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.11.7.
%H Yash Puri and Thomas Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, Journal of Integer Sequences, Vol. 4 (2001), Article 01.2.1.
%F The g.f. R[ z_ ] below (in the Mathematica field) was found by Kurt Persson (kurt(AT)math.chalmers.se) and communicated by Einar Steingrimsson (einar(AT)math.chalmers.se).
%F Using Stirling's formula in A000142, it is easy to get the asymptotic expression a(n) ~ (1/2) * (27/4)^n / sqrt(Pi*n / 3). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 07 2001
%F a(n) = Sum_{k=0..n} C(n, k)*C(2n, k). - _Paul Barry_, May 15 2003
%F G.f.: 1/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e., (in Maple notation) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - _Emeric Deutsch_, May 22 2003
%F G.f.: x*B'(x)/B(x), where B(x)+1 is the g.f. for A001764. - _Vladimir Kruchinin_, Oct 02 2015
%F a(n) ~ (1/2)*3^(1/2)*Pi^(-1/2)*n^(-1/2)*2^(-2*n)*3^(3*n)*(1 - 7/72*n^-1 + 49/10368*n^-2 + 6425/2239488*n^-3 - ...). - Joe Keane (jgk(AT)jgk.org), Nov 07 2003
%F a(n) = A006480(n)/A000984(n). - _Lior Manor_, May 04 2004
%F a(n) = Sum_{i_1=0..n, i_2=0..n} binomial(n, i_1)*binomial(n, i_2)*binomial(n, i_1+i_2). - _Benoit Cloitre_, Oct 14 2004
%F a(n) = Sum_{k=0..n} A109971(k)*3^k; a(0)=1, a(n) = Sum_{k=0..n} 3^k*C(3n-k,n-k)2k/(3n-k), n>0. - _Paul Barry_, Jan 21 2007
%F a(n) = A085478(2n,n). - _Philippe Deléham_, Sep 17 2009
%F E.g.f.: 2F2(1/3,2/3;1/2,1;27*x/4), where F(a1,a2;b1,b2;z) is a hypergeometric series. - _Emanuele Munarini_, Apr 12 2011
%F a(n) = Sum_{k=0..n} binomial(2*n+k-1,k)). - _Arkadiusz Wesolowski_, Apr 02 2012
%F G.f.: cos((1/3)*asin(sqrt(27x/4)))/sqrt(1-27x/4). - _Tom Copeland_, May 24 2012
%F G.f.: A(x) = 1 + 6*x/(G(0)-6*x) where G(k) = (2*k+2)*(2*k+1) + 3*x*(3*k+1)*(3*k+2) - 6*x*(k+1)*(2*k+1)*(3*k+4)*(3*k+5)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - _Sergei N. Gladkovskii_, Jun 30 2012
%F D-finite with recurrence: 2*n*(2*n-1)*a(n) - 3*(3*n-1)*(3*n-2)*a(n-1) = 0. - _R. J. Mathar_, Feb 05 2013
%F a(n) = (2n+1)*A001764(n). - _Johannes W. Meijer_, Aug 22 2013
%F a(n) = C(3*n-1,n-1)*C(9*n^2,2)/(3*n*C(3*n+1,3)), n>0. - _Gary Detlefs_, Jan 02 2014
%F a(n) = [x^n] 1/(1 - x)^(2*n+1). - _Ilya Gutkovskiy_, Oct 03 2017
%F a(n) = hypergeom([-2*n, -n], [1], 1). - _Peter Luschny_, Mar 19 2018
%F a(n) = Sum_{k=0..n} binomial(n, k) * binomial(2*n, n-k) = row sums of A110608. - _Michael Somos_, Jan 30 2019
%F 0 = a(n)*(-3188646*a(n+2) +7322076*a(n+3) -2805111*a(n+4) +273585*a(n+5)) +a(n+1)*(+413343*a(n+2) -1252017*a(n+3) +538344*a(n+4) -55940*a(n+5)) +a(n+2)*(-4131*a(n+2) +38733*a(n+3) -21628*a(n+4) +2528*a(n+5)) for all n in Z. - _Michael Somos_, Jan 30 2019
%F Sum_{n>=1} 1/a(n) = A229705. - _Amiram Eldar_, Nov 14 2020
%F From _Peter Bala_, Feb 20 2022: (Start)
%F The o.g.f. A(x) satisfies the differential equation (4*x - 27*x^2)*A''(x) + (2 - 54*x)*A'(x) - 6*A(x) = 0, with A(0) = 1 and A'(0) = 3.
%F Algebraic equation: (1 - A(x))*(1 + 2*A(x))^2 + 27*x*A(x)^3 = 0.
%F Sum_{n >= 1} a(n)*( x*(2*x + 3)^2/(27*(1 + x)^3) )^n = x. (End)
%F From _Vaclav Kotesovec_, May 13 2022: (Start)
%F Sum_{n>=0} a(n) / 3^(2*n) = 2*cos(Pi/9).
%F Sum_{n>=0} a(n) / (27/2)^n = (1 + sqrt(3))/2.
%F Sum_{n>=0} a(n) / 3^(3*n) = 2*cos(Pi/18) / sqrt(3).
%F In general, for k > 27/4, Sum_{n>=0} a(n)/k^n = 2*cos(arccos(1 - 27/(2*k))/6) / sqrt(4 - 27/k). (End)
%F G.f.: hypergeom([1/3, 2/3], [1/2], 27*z/4), the Gauss hypergeometric function 2F1. - _Karol A. Penson_, Dec 12 2023
%e G.f. = 1 + 3*x + 15*x^2 + 84*x^3 + 495*x^4 + 3003*x^5 + 18564*x^6 + ... - _Michael Somos_, Jan 30 2019
%p A005809:=n->binomial(3*n,n); seq(A005809(n), n=0..40); # _Wesley Ivan Hurt_, Mar 21 2014
%t R[ z_ ] := ((2-18*z + 27*z^2 + 3^(3/2)*z^(3/2)*(27*z-4)^(1/2))/2)^(1/3); f[ z_ ] := ( (R[ z ])^3 + (1-3*z)*(R[ z ])^2 + (1-6*z)*R[ z ] )/( (R[ z ])^4 + (1-6*z)*(R[ z ])^2 + (6*z-1)^2 )
%t Table[Binomial[3*n,n],{n,0,40}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 03 2011 *)
%o (Sage) [binomial(3*n,n) for n in range(0, 22)] # _Zerinvary Lajos_, Dec 16 2009
%o (Maxima) makelist(binomial(3*n,n),n,0,100); /* _Emanuele Munarini_, Apr 07 2011 */
%o (Magma) [ Binomial(3*n,n): n in [0..150] ]; // _Vincenzo Librandi_, Apr 21 2011
%o (Haskell)
%o a005809 n = a007318 (3*n) n -- _Reinhard Zumkeller_, May 06 2012
%o (PARI) a(n)=binomial(3*n,n) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Maxima)
%o B(x):=(2/sqrt(3*x))*sin((1/3)*asin(sqrt(27*x/4)))-1;
%o taylor(x*diff(B(x),x)/B(x),x,0,10); /* _Vladimir Kruchinin_, Oct 02 2015 */
%Y binomial(k*n,n): A000984 (k = 2), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
%Y Cf. A001764, A007318, A110608, A229705, A254157, A268196.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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