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%I #23 Jul 16 2020 09:11:35
%S 0,1,513,10097891,2647111616000,5170142516807540224,
%T 52103129720841632885243904,2102549272223560776918400601161728,
%U 282199388424234851655058321255905292713984,109329825340451764123791003609208862665771818418176
%N a(n) = (n!)^9 * Sum_{i=1..n} 1/i^9.
%H Seiichi Manyama, <a href="/A291507/b291507.txt">Table of n, a(n) for n = 0..75</a>
%F a(0) = 0, a(1) = 1, a(n+1) = (n^9+(n+1)^9)*a(n) - n^18*a(n-1) for n > 0.
%F a(n) ~ zeta(9) * (2*Pi)^(9/2) * n^(9*n+9/2) / exp(9*n). - _Vaclav Kotesovec_, Aug 27 2017
%F Sum_{n>=0} a(n) * x^n / (n!)^9 = polylog(9,x) / (1 - x). - _Ilya Gutkovskiy_, Jul 15 2020
%t Table[(n!)^9 * Sum[1/i^9, {i, 1, n}], {n, 0, 12}] (* _Vaclav Kotesovec_, Aug 27 2017 *)
%o (PARI) a(n) = n!^9*sum(i=1, n, 1/i^9); \\ _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), this sequence (k=9), A291508 (k=10).
%Y Column k=9 of A291556.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 25 2017
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