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%I #7 Mar 18 2019 05:27:23
%S 1,1,2,6,30,150,900,6300,52920,516600,5166000,56826000,689396400,
%T 9135126000,127891764000,1918376460000,32083427376000,547888316976000,
%U 9941031356256000,188879595768864000,3817981890122400000,80266319244951840000,1769328161437636800000
%N Expansion of e.g.f. Product_{k>=1} 1/(1 - x^(k^2)/k^2).
%F a(n) ~ 2 * n!. - _Vaclav Kotesovec_, Mar 18 2019
%t nmax = 22; CoefficientList[Series[Product[1/(1 - x^k^2/k^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Boole[IntegerQ[d^(1/2)]] d^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[IntegerQ[d^(1/2)]] d^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
%Y Cf. A007841, A319104.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 17 2019
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