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A183204 Central terms of triangle A181544.

%I A183204 #115 Sep 26 2024 07:47:40
%S A183204 1,4,48,760,13840,273504,5703096,123519792,2751843600,62659854400,
%T A183204 1451780950048,34116354472512,811208174862904,19481055861877120,
%U A183204 471822589361293680,11511531876280913760,282665135367572129040
%N A183204 Central terms of triangle A181544.
%C A183204 The g.f. for row n of triangle A181544 is (1-x)^(3n+1)*Sum_{k>=0}C(n+k-1,k)^3*x^k.
%C A183204 This sequence is s_7 in Cooper's paper. - _Jason Kimberley_, Nov 06 2012
%C A183204 Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y*z + x*y + x*z + y + z)). - _Gheorghe Coserea_, Jul 14 2016
%C A183204 This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A183204 Every prime eventually divides some term of this sequence. - _Amita Malik_, Aug 20 2017
%H A183204 Seiichi Manyama, <a href="/A183204/b183204.txt">Table of n, a(n) for n = 0..702</a> (terms 0..499 from Jason Kimberley)
%H A183204 S. Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J., December 2012, Volume 29, Issue 1, pp 163-183.
%H A183204 Shaun Cooper, Jesús Guillera, Armin Straub, and Wadim Zudilin, <a href="http://arxiv.org/abs/1604.01106">Crouching AGM, Hidden Modularity</a>, arXiv:1604.01106 [math.NT], 5-April-2016.
%H A183204 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See s7 p. 3.
%H A183204 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A183204 Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5.
%H A183204 Robert Osburn, Armin Straub, and Wadim Zudilin, <a href="https://arxiv.org/abs/1701.04098">A modular supercongruence for 6F5: an Apéry-like story</a>, arXiv:1701.04098 [math.NT], 2017.
%H A183204 Wadim Zudilin, <a href="http://arxiv.org/abs/1210.2493">A generating function of the squares of Legendre polynomials</a>, preprint arXiv:1210.2493 [math.CA], 2012.
%F A183204 a(n) = [x^n] (1-x)^(3n+1) * Sum_{k>=0} C(n+k-1,k)^3*x^k.
%F A183204 a(n) = Sum_{j = 0..n} C(n,j)^2 * C(2*j,n) * C(j+n,j). [Formula of Wadim Zudilin provided by _Jason Kimberley_, Nov 06 2012]
%F A183204 1/Pi = sqrt(7) Sum_{n>=0} (-1)^n a(n) (11895n + 1286)/22^(3n+3). [Cooper, equation (41)] - _Jason Kimberley_, Nov 06 2012
%F A183204 G.f.: sqrt((1-13*x+(1-26*x-27*x^2)^(1/2))/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2)))*hypergeom([1/12,5/12],[1],13824*x^7/(1-21*x+8*x^2+(1-8*x)*(1-26*x-27*x^2)^(1/2))^3)^2. - _Mark van Hoeij_, May 07 2013
%F A183204 a(n) ~ 3^(3*n+3/2) / (4 * (Pi*n)^(3/2)). - _Vaclav Kotesovec_, Apr 05 2015
%F A183204 G.f. A(x) satisfies 1/(1+4*x)^2 * A( x/(1+4*x)^3 ) = 1/(1+2*x)^2 * A( x^2/(1+2*x)^3 ) [see Cooper, Guillera, Straub, Zudilin]. - _Joerg Arndt_, Apr 08 2016
%F A183204 a(n) = (-1)^n*binomial(3n+1,n)* 4F3({-n,n+1,n+1,n+1};{1,1,2(n+1)}; 1). - _M. Lawrence Glasser_, May 15 2016
%F A183204 Conjecture D-finite with recurrence: n^3*a(n) - (2*n-1)*(13*n^2-13*n+4)*a(n-1) - 3*(n-1)*(3*n-4)*(3*n-2)*a(n-2) = 0. - _R. J. Mathar_, May 15 2016
%F A183204 0 = (-x^2+26*x^3+27*x^4)*y''' + (-3*x+117*x^2+162*x^3)*y'' + (-1+86*x+186*x^2)*y' + (4+24*x)*y, where y is g.f. - _Gheorghe Coserea_, Jul 14 2016
%F A183204 From _Jeremy Tan_, Mar 14 2024: (Start)
%F A183204 The conjectured D-finite recurrence can be proved by Zeilberger's algorithm.
%F A183204 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,n) * binomial(2*n-k,n) = [(w*x*y*z)^n] ((w+y)*(x+z)*(y+z)*(w+x+y+z))^n. (End)
%F A183204 a(n) = Sum_{0 <= j, k <= n} binomial(n, k)^2 * binomial(n, j)^2 * binomial(k+j, n) = Sum_{k = 0..n} binomial(n, k)^2 * A108625(n, k). - _Peter Bala_, Jul 08 2024
%F A183204 From _Peter Bala_, Sep 18 2024: (Start)
%F A183204 a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n+k, k)^3*binomial(3*n+1, n-k). Cf A245086.
%F A183204 a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n, k)*A143007(n, k) (verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package). (End)
%e A183204 Triangle A181544 begins:
%e A183204 (1);
%e A183204 1, (4), 1;
%e A183204 1, 20, (48), 20, 1;
%e A183204 1, 54, 405, (760), 405, 54, 1;
%e A183204 1, 112, 1828, 8464, (13840), 8464, 1828, 112, 1; ...
%t A183204 Table[Sum[Binomial[n,j]^2 * Binomial[2*j,n] * Binomial[j+n,j],{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Apr 05 2015 *)
%o A183204 (PARI) {a(n)=polcoeff((1-x)^(3*n+1)*sum(j=0, 2*n, binomial(n+j, j)^3*x^j), n)}
%o A183204 (Magma) P<x>:=PolynomialRing(Integers()); C:=Binomial;
%o A183204 A183204:=func<n|Coefficient((1-x)^(3*n+1)*&+[C(n+j,j)^3*x^j:j in[0..2*n]],n)>; // or directly:
%o A183204 A183204:=func<k|&+[C(k,j)^2*C(2*j,k)*C(j+k,j):j in[0..k]]>;
%o A183204 [A183204(n):n in[0..16]]; // _Jason Kimberley_, Oct 29 2012
%Y A183204 Cf. A108625, A181544, A183205, A245086.
%Y A183204 Related to diagonal of rational functions: A268545-A268555.
%Y A183204 The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)
%K A183204 nonn,easy
%O A183204 0,2
%A A183204 _Paul D. Hanna_, Dec 30 2010