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A005260 Sum_{k=0..n} C(n,k)^4.
(Formerly M2110)
19
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

REFERENCES

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

C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.

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).

Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes, http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf.

LINKS

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

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, 2013

V. Strehl, Recurrences and Legendre transform

Eric Weisstein's World of Mathematics, Binomial Sums.

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^3a(n) = 2(2n-1)(3n^2-3n+1)a(n-1) + (4n-3)(4n-4)(4n-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..infty} a(n)(4n+1)/36^n. [Cooper, equation (5)].

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

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.

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 July 4 07:29 EDT 2015. Contains 259191 sequences.