A319219 Expansion of e.g.f. Product_{k>=1} 1/(1 + x^k/(k - 1)!).
1, -1, 0, -3, 32, -105, 204, -3325, 52408, -376425, 1304180, -25766301, 659066484, -6675505837, 30765540974, -893416597515, 29169795361424, -380344619169729, 2379504317523300, -84225906785770525, 3388223174832010540, -55107296201168047221, 422923168260105913070
Offset: 0
Keywords
Programs
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Maple
seq(n!*coeff(series(mul(1/(1 + x^k/(k - 1)!),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019
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Mathematica
nmax = 22; CoefficientList[Series[Product[1/(1 + x^k/(k - 1)!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! nmax = 22; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(k (-(j - 1)!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d (-(d - 1)!)^(-k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]
Formula
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(-(j - 1)!)^k)).