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%I #12 Jan 04 2024 19:08:45
%S 1,1,5,58,1208,39476,1861372,119587224,10040970816,1067383279872,
%T 140110136642304,22256626639796352,4207858001708629248,
%U 933704296260740939520,240293228328619963492608,70992050129486593239246336,23863916105454465092261412864
%N a(0) = 1; a(n) = (n!)^2 * Sum_{k=0..n-1} a(k) / (k! * (n-k))^2.
%H Alois P. Heinz, <a href="/A336258/b336258.txt">Table of n, a(n) for n = 0..247</a>
%F a(n) = (n!)^2 * [x^n] 1 / (1 - polylog(2,x)).
%F a(n) ~ (n!)^2 / (-log(1-r) * r^n), where r = 0.76154294453204558806805187241... is the root of the equation polylog(2,r) = 1. - _Vaclav Kotesovec_, Jul 15 2020
%p b:= proc(n) option remember; `if`(n=0, 1,
%p add(b(n-i)/i^2, i=1..n))
%p end:
%p a:= n-> n!^2*b(n):
%p seq(a(n), n=0..16); # _Alois P. Heinz_, Jan 04 2024
%t a[0] = 1; a[n_] := a[n] = (n!)^2 Sum[a[k]/(k! (n - k))^2, {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[1/(1 - PolyLog[2, x]), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A007840, A074707, A102221, A336243, A336259, A336260, A336261.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 15 2020
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