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

1, 2, 66, 1460, 54850, 2031252, 86874564, 3848298792, 180295263810, 8709958973540, 433617084579316, 22071658807720392, 1145600816547477316, 60423221241495866600, 3231675487858598367240, 174928470621208572186960

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

C. Elsner, On recurrence formulas for sums involving binomial coefficients, Fib. Q., 43,1 (2005), 31-45.

R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014.

M. A. Perlstadt, Some Recurrences for Sums of Powers of Binomial Coefficients, Journal of Number Theory 27 (1987), pp. 304-309.

Formula

a(n) ~ 4*3^(-1/2)*Pi^(-5/2)*n^(-5/2)*2^(6*n).

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]

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

Sum_{n>=0} a(n) * x^n / (n!)^6 = (Sum_{n>=0} x^n / (n!)^6)^2. - Ilya Gutkovskiy, Jul 17 2020

Maple

a := n -> hypergeom([seq(-n, i=1..6)], [seq(1, i=1..5)], 1):
seq(simplify(a(n)), n=0..15); # Peter Luschny, Jul 27 2016

Mathematica

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 *)
Table[Sum[Binomial[n, k]^6, {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, May 03 2013 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)^6); \\ Michel Marcus, Mar 09 2016
(Python)
def A069865(n):
    m, g = 1, 0
    for k in range(n+1):
        g += m
        m = m*(n-k)**6//(k+1)**6
    return g # Chai Wah Wu, Oct 04 2022

Crossrefs

Sum_{k = 0..n} C(n,k)^m for m = 1..12: A000079, A000984, A000172, A005260, A005261, A069865, A182421, A182422, A182446, A182447, A342294, A342295.

Sequence in context: A159716 A157060 A154637 * A218433 A092884 A230735

Adjacent sequences: A069862 A069863 A069864 * A069866 A069867 A069868

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org), Jun 21 2002

Status

approved