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A005260 a(n) = Sum_{k = 0..n} C(n,k)^4.
(Formerly M2110)
20
1, 2, 18, 164, 1810, 21252, 263844, 3395016, 44916498, 607041380, 8345319268, 116335834056, 1640651321764, 23365271704712, 335556407724360, 4854133484555664, 70666388112940818, 1034529673001901732 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence is s_10 in Cooper's paper. - Jason Kimberley, Nov 25 2012

Diagonal of the rational function R(x,y,z,w) = 1/(1 - (w*x*y + w*x*z + w*y*z + x*y*z + w*x + y*z)). - Gheorghe Coserea, Jul 13 2016

REFERENCES

H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 79.

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

LINKS

Jason Kimberley, Table of n, a(n) for n = 0..250

F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.

W. Y. C. Chen, Q.-H. Hou, Y-P. Mu, A telescoping method for double summations, J. Comp. Appl. Math. 196 (2006) 553-566, eq (5.5)

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).

E. Delaygue, Arithmetic properties of Apery-like numbers, arXiv preprint arXiv:1310.4131 [math.NT], 2013.

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

V. Strehl, Recurrences and Legendre transform

Eric Weisstein's World of Mathematics, Binomial Sums

Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes.

FORMULA

a(n) ~ 2^(1/2)*Pi^(-3/2)*n^(-3/2)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002

n^3*a(n) = 2*(2*n - 1)*(3*n^2 - 3*n + 1)*a(n-1) + (4*n - 3)*(4*n - 4)*(4*n - 5)*a(n-2).

G.f.: 5*hypergeom([1/8, 3/8],[1], (4/5)*((1-16*x)^(1/2)+(1+4*x)^(1/2))*(-(1-16*x)^(1/2)+(1+4*x)^(1/2))^5/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2))^4)^2/(2*(1-16*x)^(1/2)+3*(1+4*x)^(1/2)). - Mark van Hoeij, Oct 29 2011.

1/Pi = sqrt(15)/18 * Sum_{n >= 0} a(n)*(4*n + 1)/36^n [Cooper, equation (5)],

= sqrt(15)/18 * Sum_{n >= 0} a(n)*A016813(n)/A009980(n). - Jason Kimberley, Nov 26 2012

0 = (-x^2 + 12*x^3 + 64*x^4)*y''' + (-3*x + 54*x^2 + 384*x^3)*y'' + (-1 + 40*x + 444*x^2)*y' + (2 + 60*x)*y, where y is g.f. - Gheorghe Coserea, Jul 13 2016

For r a nonnegative integer, Sum_{k = r..n} C(k,r)^4*C(n,k)^4 = C(n,r)^4*a(n-r), where we take a(n) = 0 for n < 0. - Peter Bala, Jul 27 2016

a(n) = hypergeom([-n, -n, -n, -n],[1, 1, 1],1). - Peter Luschny, Jul 27 2016

EXAMPLE

G.f. = 1 + 2*x + 18*x^2 + 164*x^3 + 1810*x^4 + 21252*x^5 + 263844*x^6 + ...

MAPLE

A005260 := proc(n)

        add( (binomial(n, k))^4, k=0..n) ;

end proc:

seq(A005260(n), n=0..10) ; # R. J. Mathar, Nov 19 2012

MATHEMATICA

Table[Sum[Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 09 2014 *)

PROG

(PARI) {a(n) = sum(k=0, n, binomial(n, k)^4)};

CROSSREFS

Cf. A000172, A096192.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A037518 A037721 A245998 * A183250 A037728 A037623

Adjacent sequences:  A005257 A005258 A005259 * A005261 A005262 A005263

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Michael Somos, Aug 09 2002

Minor edits by Vaclav Kotesovec, Aug 28 2014

STATUS

approved

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Last modified September 25 09:30 EDT 2016. Contains 276529 sequences.