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%I #23 Jul 15 2020 11:36:28
%S 0,1,257,1686433,110523752704,43173450975314176,
%T 72514862031522895036416,418033821374598847702425993216,
%U 7013444132843374500928464765799366656,301905779820559925981495987360836056017534976
%N a(n) = (n!)^8 * Sum_{i=1..n} 1/i^8.
%H Seiichi Manyama, <a href="/A291506/b291506.txt">Table of n, a(n) for n = 0..83</a>
%F a(0) = 0, a(1) = 1, a(n+1) = (n^8+(n+1)^8)*a(n) - n^16*a(n-1) for n > 0.
%F a(n) ~ 8 * Pi^12 * n^(8*n+4) / (4725 * exp(8*n)). - _Vaclav Kotesovec_, Aug 27 2017
%F Sum_{n>=0} a(n) * x^n / (n!)^8 = polylog(8,x) / (1 - x). - _Ilya Gutkovskiy_, Jul 15 2020
%t Table[(n!)^8 * Sum[1/i^8, {i, 1, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Aug 27 2017 *)
%o (PARI) a(n) = n!^8*sum(i=1, n, 1/i^8); \\ _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), this sequence (k=8), A291507 (k=9), A291508 (k=10).
%Y Column k=8 of A291556.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 25 2017
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