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A002897 a(n) = C(2*n,n)^3.
(Formerly M4580 N1952)
25
1, 8, 216, 8000, 343000, 16003008, 788889024, 40424237568, 2131746903000, 114933031928000, 6306605327953216, 351047164190381568, 19774031697705428416, 1125058699232216000000, 64561313052442296000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Diagonal of the rational function R(x,y,z) = 1/(1 - (w*x*y + w*z + x + y + z)). - Gheorghe Coserea, Jul 14 2016

REFERENCES

C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.

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).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.

FORMULA

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

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

G.f.: hypergeom([1/4,1/4],[1],64*x)^2. - Mark van Hoeij, Nov 17 2011

n^3*a(n) - 8*(2*n - 1)^3*a(n-1) = 0. - R. J. Mathar, Mar 08 2013

From Peter Bala, Jul 12 2016: (Start)

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)

a(n) ~ 64^n/(Pi*n)^(3/2). - Ilya Gutkovskiy, Jul 13 2016

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

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

MATHEMATICA

a[n_]:= Coefficient[ Series[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64x], {x, 0, n}], x, n]

PROG

(PARI) {a(n)= binomial(2*n, n)^3} /* Michael Somos, Jan 31 2007 */

(Sage) [binomial(2*n, n)**3 for n in xrange(0, 17)] # Zerinvary Lajos, Apr 21 2009

(MAGMA) [Binomial(2*n, n)^3: n in [0..20]]; // Vincenzo Librandi, Nov 18 2011

CROSSREFS

Cf. A000897, A002894, A006480, A008977, A186420, A188662.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A271400 A123057 A009072 * A024289 A009106 A000442

Adjacent sequences:  A002894 A002895 A002896 * A002898 A002899 A002900

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified July 29 03:28 EDT 2016. Contains 275161 sequences.