A291508 - OEIS

%I #21 Jul 16 2020 09:11:41

%S 0,1,1025,60526249,63466432537600,619789443653380965376,

%T 37476298202061058687475122176,10586126703664512292193022557971021824,

%U 11366767006463449393869821987386636472445566976,39633465899293694663690352980684333029782095493517541376

%N a(n) = (n!)^10 * Sum_{i=1..n} 1/i^10.

%H Seiichi Manyama, <a href="/A291508/b291508.txt">Table of n, a(n) for n = 0..69</a>

%F a(0) = 0, a(1) = 1, a(n+1) = (n^10+(n+1)^10)*a(n) - n^20*a(n-1) for n > 0.

%F a(n) ~ 32 * Pi^15 * n^(10*n+5) / (93555 * exp(10*n)). - _Vaclav Kotesovec_, Aug 27 2017

%F Sum_{n>=0} a(n) * x^n / (n!)^10 = polylog(10,x) / (1 - x). - _Ilya Gutkovskiy_, Jul 15 2020

%t Table[(n!)^10 * Sum[1/i^10, {i, 1, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Aug 27 2017 *)

%o (PARI) a(n) = n!^10*sum(i=1, n, 1/i^10); \\ _Michel Marcus_, Aug 26 2017

%Y Cf. A000254 (k=1), A001819 (k=2), A066989 (k=3), A099827 (k=5), A291456 (k=6), A291505 (k=7), A291506 (k=8), A291507 (k=9), this sequence (k=10).

%Y Column k=10 of A291556.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 25 2017