A002897 - OEIS

%I M4580 N1952 #117 Oct 18 2024 11:42:44

%S 1,8,216,8000,343000,16003008,788889024,40424237568,2131746903000,

%T 114933031928000,6306605327953216,351047164190381568,

%U 19774031697705428416,1125058699232216000000,64561313052442296000000

%N a(n) = binomial(2n,n)^3.

%C Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*z + x + y + z)). - _Gheorghe Coserea_, Jul 14 2016

%C Conjecture: The g.f. is also the diagonal of the rational function 1/(1 - (x + y)*(1 - 4*z*t) - z - t) = 1/det(I - M*diag(x, y, z, t)), I the 4 x 4 unit matrix and M the 4 x 4 matrix [1, 1, 1, 1; 1, 1, 1, 1; 1, 1, 1, -1; 1 , 1, -1, 1]. If true, then a(n) = [(x*y*z)^n] (1 + x + y + z)^(2*n)*(1 + x + y - z)^n*(1 + x - y + z)^n. - _Peter Bala_, Apr 10 2022

%D S. Ramanujan, Modular Equations and Approximations to pi, pp. 23-39 of Collected Papers of Srinivasa Ramanujan, Ed. G. H. Hardy et al., AMS Chelsea 2000. See page 36, equation (25).

%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).

%H Vincenzo Librandi, <a href="/A002897/b002897.txt">Table of n, a(n) for n = 0..100</a>

%H David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, <a href="http://arxiv.org/abs/0801.0891">Elliptic integral evaluations of Bessel moments</a>, arXiv:0801.0891 [hep-th], 2008.

%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.

%H 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 Yen Lee Loh, <a href="https://arxiv.org/abs/1706.03083">A general method for calculating lattice Green functions on the branch cut</a>, arXiv:1706.03083 [math-ph], 2017.

%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.

%F Expansion of (K(k)/(Pi/2))^2 in powers of (kk'/4)^2, where K(k) is the complete elliptic integral of the first kind evaluated at modulus k. - _Michael Somos_, Jan 31 2007

%F G.f.: F(1/2, 1/2, 1/2; 1, 1; 64x) where F() is a hypergeometric function. - _Michael Somos_, Jan 31 2007

%F G.f.: hypergeom([1/4,1/4],[1],64*x)^2. - _Mark van Hoeij_, Nov 17 2011

%F D-finite with recurrence n^3*a(n) - 8*(2*n - 1)^3*a(n-1) = 0. - _R. J. Mathar_, Mar 08 2013

%F From _Peter Bala_, Jul 12 2016: (Start)

%F a(n) = binomial(2*n,n)^3 = ( [x^n](1 + x)^(2*n) )^3 = [x^n](F(x)^(8*n)), where F(x) = 1 + x + 6*x^2 + 111*x^3 + 2806*x^4 + 84456*x^5 + 2832589*x^6 + 102290342*x^7 + ... appears to have integer coefficients. For similar results see A000897, A002894, A006480, A008977, A186420 and A188662. (End)

%F a(n) ~ 64^n/(Pi*n)^(3/2). - _Ilya Gutkovskiy_, Jul 13 2016

%F 0 = (-x^2 + 64*x^3)*y''' + (-3*x + 288*x^2)*y'' + (-1 + 208*x)*y' + 8*y, where y is g.f. - _Gheorghe Coserea_, Jul 14 2016

%F a(n) = Sum_{k = 0..n} (2*n + k)!/(k!^3*(n - k)!^2). Cf. A001850(n) = Sum_{k = 0..n} (n + k)!/(k!^2*(n - k)!). - _Peter Bala_, Jul 27 2016

%F It appears that a(n) is the coefficient of (x*y*z)^(2*n) in the expansion of (1 + x*y + x*z - y*z)^(2*n) * (1 + x*y - x*z + y*z)^(2*n) * (1 - x*y + x*z + y*z)^(2*n).  Cf. A000172. - _Peter Bala_, Sep 21 2021

%F From _Peter Bala_, Sep 24 2022: (Start)

%F a(n) = Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)*binomial(2*n+k,n).

%F a(n) = the coefficient of (x*y*z*t^2)^n in the expansion of 1/(1 - x - y)*(1 - z - t) - x*y*z*t) (a(n) = A(n,n,n,2*n) in the notation of Straub, Theorem 1.2). (End)

%F a(n) = (8/5) * Sum_{k = 0..n} binomial(n,k)^2*binomial(n+k,k)*binomial(2*n+k-1,n) for n >= 1. - _Peter Bala_, Jul 09 2024

%F a(n) = Sum_{k = 0..n} binomial(n, k)^2 * A108625(2*n, k). Cf. A183204. - _Peter Bala_, Oct 12 2024

%F From _Peter Bala_, Oct 16 2024: (Start)

%F a(n) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k)*binomial(2*n+k, k)*A108625(n, k) = 8 * Sum_{k = 0..n} (-1)^(n+k+1) * binomial(n-1, k)*binomial(2*n+k-1, k)*A108625(n, k) = (8/5) * Sum_{k = 0..n} (-1)^(n+k) * binomial(n, k)*binomial(2*n+k-1, k)*A108625(n, k) for n >= 1. Cf. A176285. (End)

%t a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64x], {x, 0, n}];

%t Table[Binomial[2n,n]^3,{n,0,20}] (* _Harvey P. Dale_, Dec 06 2017 *)

%o (PARI) {a(n) = binomial(2*n, n)^3}; /* _Michael Somos_, Jan 31 2007 */

%o (Sage) [binomial(2*n, n)**3 for n in range(21)] # _Zerinvary Lajos_, Apr 21 2009

%o (Magma) [Binomial(2*n, n)^3: n in [0..20]]; // _Vincenzo Librandi_, Nov 18 2011

%Y Cf. A000897, A002894, A006480, A008977, A108625, A176285, A183204, A186420, A188662.

%Y Related to diagonal of rational functions: A268545-A268555.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_