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%I A002897 M4580 N1952
%S A002897 1,8,216,8000,343000,16003008,788889024,40424237568,2131746903000,
%T A002897 114933031928000,6306605327953216,351047164190381568,
%U A002897 19774031697705428416,1125058699232216000000,64561313052442296000000
%N A002897 a(n) = binomial(2n,n)^3.
%C A002897 Diagonal of the rational function R(x,y,z) = 1/(1 - (w*x*y + w*z + x + y + z)). - _Gheorghe Coserea_, Jul 14 2016
%D A002897 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 A002897 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002897 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002897 Vincenzo Librandi, <a href="/A002897/b002897.txt">Table of n, a(n) for n = 0..100</a>
%H A002897 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 A002897 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 A002897 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.
%F A002897 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 A002897 G.f.: F(1/2, 1/2, 1/2; 1, 1; 64x) where F() is a hypergeometric function. - _Michael Somos_, Jan 31 2007
%F A002897 G.f.: hypergeom([1/4,1/4],[1],64*x)^2. - _Mark van Hoeij_, Nov 17 2011
%F A002897 D-finite with recurrence n^3*a(n) - 8*(2*n - 1)^3*a(n-1) = 0. - _R. J. Mathar_, Mar 08 2013
%F A002897 From _Peter Bala_, Jul 12 2016: (Start)
%F A002897 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 A002897 a(n) ~ 64^n/(Pi*n)^(3/2). - _Ilya Gutkovskiy_, Jul 13 2016
%F A002897 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 A002897 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
%t A002897 a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64x], {x, 0, n}];
%t A002897 Table[Binomial[2n,n]^3,{n,0,20}] (* _Harvey P. Dale_, Dec 06 2017 *)
%o A002897 (PARI) {a(n) = binomial(2*n, n)^3}; /* _Michael Somos_, Jan 31 2007 */
%o A002897 (Sage) [binomial(2*n, n)**3 for n in range(21)] # _Zerinvary Lajos_, Apr 21 2009
%o A002897 (MAGMA) [Binomial(2*n, n)^3: n in [0..20]]; // _Vincenzo Librandi_, Nov 18 2011
%Y A002897 Cf. A000897, A002894, A006480, A008977, A186420, A188662.
%Y A002897 Related to diagonal of rational functions: A268545-A268555.
%K A002897 nonn,easy
%O A002897 0,2
%A A002897 _N. J. A. Sloane_, _Simon Plouffe_

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