cp's OEIS Frontend

A069865 a(n) = Sum_{k = 0..n} C(n,k)^6.

%I A069865 #49 Jun 27 2025 18:53:15
%S A069865 1,2,66,1460,54850,2031252,86874564,3848298792,180295263810,
%T A069865 8709958973540,433617084579316,22071658807720392,1145600816547477316,
%U A069865 60423221241495866600,3231675487858598367240,174928470621208572186960
%N A069865 a(n) = Sum_{k = 0..n} C(n,k)^6.
%H A069865 Vincenzo Librandi, <a href="/A069865/b069865.txt">Table of n, a(n) for n = 0..200</a>
%H A069865 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 A069865 R. Mestrovic, <a href="http://arxiv.org/abs/1409.3820">Lucas' theorem: its generalizations, extensions and applications (1878--2014)</a>, arXiv preprint arXiv:1409.3820 [math.NT], 2014.
%H A069865 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 A069865 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 A069865 a(n) ~ 4*3^(-1/2)*Pi^(-5/2)*n^(-5/2)*2^(6*n).
%F A069865 Recurrence (M. A. Perlstadt, 1987): 24*(6*n - 7)*(2*n - 1)*(6*n - 5)*(91*n^3 + 91*n^2 + 35*n + 5)*(n - 1)^3*a(n-2) - (153881*n^9 - 307762*n^8 + 185311*n^7 + 2960*n^6 - 31631*n^5 - 88*n^4 + 5239*n^3 - 610*n^2 - 440*n + 100)*a(n-1) - n*(3458*n^8 + 1729*n^7 - 2947*n^6 - 2295*n^5 + 901*n^4 + 1190*n^3 + 52*n^2 - 228*n - 60)*a(n) + n*(91*n^3 - 182*n^2 + 126*n - 30)*(n + 1)^5*a(n+1) = 0. [_Vaclav Kotesovec_, Apr 27 2012]
%F A069865 For r a nonnegative integer, Sum_{k = r..n} C(k,r)^6*C(n,k)^6 = C(n,r)^6*a(n-r), where we take a(n) = 0 for n < 0. - _Peter Bala_, Jul 27 2016
%F A069865 Sum_{n>=0} a(n) * x^n / (n!)^6 = (Sum_{n>=0} x^n / (n!)^6)^2. - _Ilya Gutkovskiy_, Jul 17 2020
%F A069865 From _Peter Bala_, Nov 01 2024: (Start)
%F A069865 For n >= 1, a(n) = 2 * Sum_{k = 0..n-1} binomial(n, k)^5 * binomial(n-1, k).
%F A069865 For n >= 1, a(n) = 2 * hypergeom([-n, -n, -n, -n, -n, -n + 1], [1, 1, 1, 1, 1], 1).
%F A069865 (End)
%p A069865 a := n -> hypergeom([seq(-n, i=1..6)],[seq(1, i=1..5)],1):
%p A069865 seq(simplify(a(n)),n=0..15); # _Peter Luschny_, Jul 27 2016
%t A069865 RecurrenceTable[{24(6n-7)(2n-1)(6n-5)(91n^3 + 91n^2 + 35n + 5)(n-1)^3*a[n-2] -(153881n^9-307762n^8 + 185311n^7 + 2960n^6-31631n^5-88n^4 + 5239n^3-610n^2-440n + 100)*a[n-1] -n(3458n^8 + 1729n^7-2947n^6-2295n^5 + 901n^4 + 1190n^3 + 52n^2-228n-60)*a[n] + n(91n^3-182n^2 + 126n-30)(n + 1)^5*a[n + 1]==0, a[0]==1,a[1]==2,a[2]==66},a,{n,0,25}] (* _Vaclav Kotesovec_, Apr 27 2012 *)
%t A069865 Table[Sum[Binomial[n, k]^6, {k, 0, n}], {n, 0, 25}] (* _Vincenzo Librandi_, May 03 2013 *)
%o A069865 (PARI) a(n) = sum(k=0, n, binomial(n, k)^6); \\ _Michel Marcus_, Mar 09 2016
%o A069865 (Python)
%o A069865 def A069865(n):
%o A069865     m, g = 1, 0
%o A069865     for k in range(n+1):
%o A069865         g += m
%o A069865         m = m*(n-k)**6//(k+1)**6
%o A069865     return g # _Chai Wah Wu_, Oct 04 2022
%Y A069865 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 A069865 nonn,easy
%O A069865 0,2
%A A069865 Joe Keane (jgk(AT)jgk.org), Jun 21 2002