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%I #28 Mar 27 2021 08:15:21
%S 1,2,130,4376,312706,20156252,1622278624,132282417920,11716609750402,
%T 1067553850832372,101275413131018380,9844149854624122160,
%U 980597565209377223200,99518148302583383833280,10272819477206557916630080,1075608762378043981968997376
%N a(n) = Sum_{k = 0..n} C(n,k)^7.
%H Vincenzo Librandi, <a href="/A182421/b182421.txt">Table of n, a(n) for n = 0..200</a>
%H Vaclav Kotesovec, <a href="/A182421/a182421_1.txt">Recurrence (of order 4)</a>
%H 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.
%F Asymptotic (p = 7): a(n) ~ 2^(p*n)/sqrt(p)*(2/(Pi*n))^((p - 1)/2)*( 1 - (p - 1)^2/(4*p*n) + O(1/n^2) ).
%F For r a nonnegative integer, Sum_{k = r..n} C(k,r)^7*C(n,k)^7 = C(n,r)^7*a(n-r), where we take a(n) = 0 for n < 0. - _Peter Bala_, Jul 27 2016
%F Sum_{n>=0} a(n) * x^n / (n!)^7 = (Sum_{n>=0} x^n / (n!)^7)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%p a := n -> hypergeom([seq(-n, i=1..7)],[seq(1, i=1..6)],-1):
%p seq(simplify(a(n)),n=0..15); # _Peter Luschny_, Jul 27 2016
%t Table[Total[Binomial[n, Range[0, n]]^7], {n, 0, 20}] (* _T. D. Noe_, Apr 28 2012 *)
%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^7); \\ _Michel Marcus_, Jul 17 2020
%Y 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 nonn,easy
%O 0,2
%A _Vaclav Kotesovec_, Apr 28 2012
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