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 A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4. (Formerly M2110) 52

%I M2110

%S 1,2,18,164,1810,21252,263844,3395016,44916498,607041380,8345319268,

%T 116335834056,1640651321764,23365271704712,335556407724360,

%U 4854133484555664,70666388112940818,1034529673001901732

%N a(n) = Sum_{k = 0..n} binomial(n,k)^4.

%C This sequence is s_10 in Cooper's paper. - _Jason Kimberley_, Nov 25 2012

%C 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

%C This is one of the Apery-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017

%C Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017

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

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

%H Seiichi Manyama, <a href="/A005260/b005260.txt">Table of n, a(n) for n = 0..834</a> (terms 0..250 from Jason Kimberley)

%H B. Adamczewski, J. P. Bell, E. Delaygue, <a href="https://arxiv.org/abs/1603.04187">Algebraic independence of G-functions and congruences "a la Lucas"</a>, arXiv preprint arXiv:1603.04187 [math.NT], 2016.

%H F. Beukers, <a href="http://dx.doi.org/10.1016/0022-314X(87)90025-4">Another congruence for the Apéry numbers</a>, J. Number Theory 25 (1987), no. 2, 201-210.

%H W. Y. C. Chen, Q.-H. Hou, Y-P. Mu, <a href="http://dx.doi.org/10.1016/j.cam.2005.10.010">A telescoping method for double summations</a>, J. Comp. Appl. Math. 196 (2006) 553-566, eq (5.5)

%H S. Cooper, <a href="http://dx.doi.org/10.1007/s11139-011-9357-3">Sporadic sequences, modular forms and new series for 1/pi</a>, Ramanujan J. (2012).

%H M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>

%H E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apery-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.

%H C. Elsner, <a href="http://www.fq.math.ca/Papers1/43-1/paper43-1-5.pdf">On recurrence formulas for sums involving binomial coefficients</a>, Fib. Q., 43,1 (2005), 31-45.

%H Amita Malik and Armin Straub, <a href="https://doi.org/10.1007/s40993-016-0036-8">Divisibility properties of sporadic Apéry-like numbers</a>, Research in Number Theory, 2016, 2:5

%H Robert Osburn, Armin Straub, Wadim Zudilin, <a href="https://arxiv.org/abs/1701.04098">A modular supercongruence for 6F5: an Apéry-like story</a>, arXiv:1701.04098 [math.NT], 2017.

%H V. Strehl, <a href="http://www.mat.univie.ac.at/~slc/opapers/s29strehl.html">Recurrences and Legendre transform</a>, Séminaire Lotharingien de Combinatoire, B29b (1992), 22 pp.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>

%H Mark C. Wilson, <a href="http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf">Diagonal asymptotics for products of combinatorial classes</a>.

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

%F 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).

%F 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.

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

%F = sqrt(15)/18 * Sum_{n >= 0} a(n)*A016813(n)/A009980(n). - _Jason Kimberley_, Nov 26 2012

%F 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

%F 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

%F a(n) = hypergeom([-n, -n, -n, -n],[1, 1, 1],1). - _Peter Luschny_, Jul 27 2016

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

%p A005260 := proc(n)

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

%p end proc:

%p seq(A005260(n),n=0..10) ; # _R. J. Mathar_, Nov 19 2012

%t Table[Sum[Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 09 2014 *)

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

%Y Cf. A000172, A096192.

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

%Y The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692,A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Edited by _Michael Somos_, Aug 09 2002

%E Minor edits by _Vaclav Kotesovec_, Aug 28 2014

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Last modified November 17 12:09 EST 2018. Contains 317276 sequences. (Running on oeis4.)