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%I M2156
%S 1,2,34,488,9826,206252,4734304,113245568,2816649826,72001228052,
%T 1883210876284,50168588906768,1357245464138656,37198352117916992,
%U 1030920212982957184,28847760730478655488,814066783370083977826
%N Sum C(n,k)^5, k = 0 . . n.
%D C. Elsner, On recurrence formulae for sums involving binomial coefficients, Fib. Q., 43 (No. 1, 2005), 31-45.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.
%H Vincenzo Librandi, <a href="/A005261/b005261.txt">Table of n, a(n) for n = 0..200</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BinomialSums.html">Binomial sums.</a>
%F a(n) ~ 4*5^(-1/2)*pi^-2*n^-2*2^(5*n) - Joe Keane (jgk(AT)jgk.org), Jun 21 2002
%F 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]
%t 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 (* or directly *)
%t Table[Sum[Binomial[n,k]^5,{k,0,n}],{n,0,25}] (* From _Vaclav Kotesovec_, Apr 27 2012 *)
%Y Cf. A000079, A000984, A000172, A005260, A069865
%K nonn
%O 0,2
%A _N. J. A. Sloane_.
%E More terms from _Matthew Conroy_, Mar 16 2006
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