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

%I A005260 M2110 #195 Apr 19 2025 17:50:37
%S A005260 1,2,18,164,1810,21252,263844,3395016,44916498,607041380,8345319268,
%T A005260 116335834056,1640651321764,23365271704712,335556407724360,
%U A005260 4854133484555664,70666388112940818,1034529673001901732,15220552520052960516,224929755893153896200,3337324864503769353060
%N A005260 a(n) = Sum_{k = 0..n} binomial(n,k)^4.
%C A005260 This sequence is s_10 in Cooper's paper. - _Jason Kimberley_, Nov 25 2012
%C A005260 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 A005260 This is one of the Apéry-like sequences - see Cross-references. - _Hugo Pfoertner_, Aug 06 2017
%C A005260 Every prime eventually divides some term of this sequence. - _Amita Malik_, Aug 20 2017
%C A005260 Two walkers, A and B, stand on the South-West and North-East corners of an n X n grid, respectively. A walks by either North or East steps while B walks by either South or West steps. Sequence values a(n) < binomial(2*n,n)^2 count the simultaneous walks where A and B meet after exactly n steps and change places after 2*n steps. - _Bradley Klee_, Apr 01 2019
%C A005260 a(n) is the constant term in the expansion of ((1 + x) * (1 + y) * (1 + z) + (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - _Seiichi Manyama_, Oct 27 2019
%D A005260 H. W. Gould, Combinatorial Identities, Morgantown, 1972, (X.14), p. 79.
%D A005260 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005260 Seiichi Manyama, <a href="/A005260/b005260.txt">Table of n, a(n) for n = 0..834</a> (terms 0..250 from Jason Kimberley)
%H A005260 B. Adamczewski, J. P. Bell, and 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 A005260 Hacene Belbachir and Yassine Otmani, <a href="https://arxiv.org/abs/2012.02563">A Strehl Version of Fourth Franel Sequence</a>, arXiv:2012.02563 [math.CO], 2020.
%H A005260 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 A005260 W. Y. C. Chen, Q.-H. Hou, and 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 A005260 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 A005260 M. Coster, <a href="/A001850/a001850_1.pdf">Email, Nov 1990</a>
%H A005260 E. Delaygue, <a href="http://arxiv.org/abs/1310.4131">Arithmetic properties of Apéry-like numbers</a>, arXiv preprint arXiv:1310.4131 [math.NT], 2013.
%H A005260 C. Elsner, <a href="https://web.archive.org/web/2024*/https://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 A005260 Ofir Gorodetsky, <a href="https://arxiv.org/abs/2102.11839">New representations for all sporadic Apéry-like sequences, with applications to congruences</a>, arXiv:2102.11839 [math.NT], 2021. See s10 p. 3.
%H A005260 Darij Grinberg, <a href="http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf">Introduction to Modern Algebra</a> (UMN Spring 2019 Math 4281 Notes), University of Minnesota (2019).
%H A005260 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%H A005260 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 A005260 Robert Osburn, Armin Straub, and 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 A005260 M. A. Perlstadt, <a href="http://dx.doi.org/10.1016/0022-314X(87)90069-2">Some Recurrences for Sums of Powers of Binomial Coefficients</a>, Journal of Number Theory 27 (1987), pp. 304-309.
%H A005260 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 A005260 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/191o.pdf">Open conjectures on congruences</a>, Nanjing Univ. J. Math. Biquarterly 36(2019), no.1, 1-99. (Cf. Conjectures 49-51.)
%H A005260 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial Sums</a>
%H A005260 Mark C. Wilson, <a href="http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf">Diagonal asymptotics for products of combinatorial classes</a>, preprint of Combinatorics, Probability and Computing, 24(1), 2015, 354-372.
%H A005260 Jin Yuan, Zhi-Juan Lu, and Asmus L. Schmidt, <a href="https://dx.doi.org/10.1016/j.jnt.2008.03.011">On recurrences for sums of powers of binomial coefficients</a>, J. Numb. Theory 128 (2008) 2784-2794.
%F A005260 a(n) ~ 2^(1/2)*Pi^(-3/2)*n^(-3/2)*2^(4*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
%F A005260 D-finite with recurrence: 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). [Yuan]
%F A005260 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 A005260 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
%F A005260 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 A005260 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 A005260 a(n) = hypergeom([-n, -n, -n, -n], [1, 1, 1], 1). - _Peter Luschny_, Jul 27 2016
%F A005260 Sum_{n>=0} a(n) * x^n / (n!)^4 = (Sum_{n>=0} x^n / (n!)^4)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%F A005260 a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k)*C(2k,k)*C(2n-2k,n-k)*(-1)^(n-k). This can be proved via the Zeilberger algorithm. - _Zhi-Wei Sun_, Aug 23 2020
%F A005260 a(n) = (-1)^n*binomial(2*n, n)*hypergeom([1/2, -n, -n, n + 1], [1, 1, 1/2 - n], 1). - _Peter Luschny_, Aug 24 2020
%F A005260 a(n) = Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,n)*binomial(2*n-k,n) [Theorem 1 in Belbachir and Otmani]. - _Michel Marcus_, Dec 06 2020
%F A005260 a(n) = [x^n] (1 - x)^(2*n) P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. See Gould, p. 66. This formula is equivalent to the binomial sum identity of _Zhi-Wei Sun_ given above. - _Peter Bala_, Mar 24 2022
%F A005260 From _Peter Bala_, Oct 31 2024: (Start)
%F A005260 For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^3 * binomial(n-1, k).
%F A005260 For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n + 1], [1, 1, 1], 1). (End)
%F A005260 G.f.: Sum_{k>=0} Sum_{l=0..p*k} Sum_{m=0..l} (-1)^m*binomial(p*k+1,m)*binomial(l+k-m,k)^p*x^(l+k)/(1-x)^(p*k+1), where p = 4. - _Miles Wilson_, Apr 12 2025
%e A005260 G.f. = 1 + 2*x + 18*x^2 + 164*x^3 + 1810*x^4 + 21252*x^5 + 263844*x^6 + ...
%p A005260 A005260 := proc(n)
%p A005260         add( (binomial(n,k))^4,k=0..n) ;
%p A005260 end proc:
%p A005260 seq(A005260(n),n=0..10) ; # _R. J. Mathar_, Nov 19 2012
%t A005260 Table[Sum[Binomial[n, k]^4, {k, 0, n}], {n, 0, 20}] (* _Wesley Ivan Hurt_, Mar 09 2014 *)
%o A005260 (PARI) {a(n) = sum(k=0, n, binomial(n, k)^4)};
%o A005260 (Python)
%o A005260 def A005260(n):
%o A005260     m, g = 1, 0
%o A005260     for k in range(n+1):
%o A005260         g += m
%o A005260         m = m*(n-k)**4//(k+1)**4
%o A005260     return g # _Chai Wah Wu_, Oct 04 2022
%Y A005260 Column k=4 of A309010.
%Y A005260 Cf. A000172, A096192, A328725.
%Y A005260 Related to diagonal of rational functions: A268545-A268555.
%Y A005260 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.)
%Y A005260 Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
%Y A005260 Row sums of A202750.
%K A005260 nonn,easy
%O A005260 0,2
%A A005260 _N. J. A. Sloane_
%E A005260 Edited by _Michael Somos_, Aug 09 2002
%E A005260 Minor edits by _Vaclav Kotesovec_, Aug 28 2014