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A005261 a(n) = Sum_{k = 0..n} C(n,k)^5.

%I A005261 M2156 #85 Jun 27 2025 19:06:12
%S A005261 1,2,34,488,9826,206252,4734304,113245568,2816649826,72001228052,
%T A005261 1883210876284,50168588906768,1357245464138656,37198352117916992,
%U A005261 1030920212982957184,28847760730478655488,814066783370083977826
%N A005261 a(n) = Sum_{k = 0..n} C(n,k)^5.
%C A005261 a(n) is the constant term in the expansion of ((1 + w) * (1 + x) * (1 + y) * (1 + z) + (1 + 1/w) * (1 + 1/x) * (1 + 1/y) * (1 + 1/z))^n. - _Seiichi Manyama_, Oct 27 2019
%D A005261 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005261 Vincenzo Librandi, <a href="/A005261/b005261.txt">Table of n, a(n) for n = 0..200</a>
%H A005261 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 A005261 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 A005261 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialSums.html">Binomial sums</a>.
%H A005261 Mark C. Wilson, <a href="http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf">Diagonal asymptotics for products of combinatorial classes</a>.
%H A005261 Jin Yuan, Zhi-Juan Lu, 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 A005261 a(n) ~ 4*5^(-1/2)*Pi^-2*n^-2*2^(5*n). - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
%F A005261 Recurrence (M. A. Perlstadt, 1987): 32*(55*n^2 + 33*n + 6)*(n - 1)^4*a(n-2) - (19415*n^6 - 27181*n^5 + 7453*n^4 + 3289*n^3 - 956*n^2 - 276*n + 96)*a(n-1) - (1155*n^6 + 693*n^5 - 732*n^4 - 715*n^3 + 45*n^2 + 210*n + 56)*a(n) + (55*n^2 - 77*n + 28)*(n + 1)^4*a(n+1) = 0. [_Vaclav Kotesovec_, Apr 27 2012]
%F A005261 For r a nonnegative integer, Sum_{k = r..n} C(k,r)^5*C(n,k)^5 = C(n,r)^5*a(n-r), where we take a(n) = 0 for n < 0. - _Peter Bala_, Jul 27 2016
%F A005261 Sum_{n>=0} a(n) * x^n / (n!)^5 = (Sum_{n>=0} x^n / (n!)^5)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%F A005261 From _Peter Bala_, Nov 01 2024: (Start)
%F A005261 For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^4 * binomial(n-1, k).
%F A005261 For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n, -n + 1], [1, 1, 1, 1], -1). (End)
%p A005261 a := n -> hypergeom([seq(-n, i=1..5)],[seq(1, i=1..4)], -1):
%p A005261 seq(simplify(a(n)),n=0..16); # _Peter Luschny_, Jul 27 2016
%t A005261 RecurrenceTable[{32*(55n^2+33n+6)*(n-1)^4*a[n-2]-(19415n^6-27181n^5+7453n^4+3289n^3-956n^2-276n+96)*a[n-1]-(1155n^6+693n^5-732n^4-715n^3+45n^2+210n+56)*a[n]+(55n^2-77n+28)*(n+1)^4*a[n+1]==0,a[0]==1,a[1]==2,a[2]==34},a,{n,0,25}]
%t A005261 (* or directly *)
%t A005261 Table[Sum[Binomial[n,k]^5,{k,0,n}],{n,0,25}] (* _Vaclav Kotesovec_, Apr 27 2012 *)
%o A005261 (PARI) a(n) = sum(k=0, n, binomial(n, k)^5); \\ _Michel Marcus_, Mar 09 2016
%o A005261 (Python)
%o A005261 def A005261(n):
%o A005261     m, g = 1, 0
%o A005261     for k in range(n+1):
%o A005261         g += m
%o A005261         m = m*(n-k)**5//(k+1)**5
%o A005261     return g # _Chai Wah Wu_, Oct 04 2022
%Y A005261 Column k=5 of A309010.
%Y A005261 Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.
%K A005261 nonn,easy
%O A005261 0,2
%A A005261 _N. J. A. Sloane_
%E A005261 More terms from _Matthew Conroy_, Mar 16 2006