A380171 Numerators of coefficients in expansion of exp(-1 + 1 / Product_{k>=1} (1 - x^k)).

1, 1, 5, 31, 265, 2621, 31621, 85319, 6574961, 22334789, 2092318021, 42552808871, 187499032037, 22150499622421, 22390616112461, 15039597200385451, 428293292251548001, 103005657594642373, 407547173842501629061, 2708181047424714819491, 36245898714951203790797

Offset: 0

Author

Ilya Gutkovskiy, Jan 14 2025

Keywords

• nonn
• frac

Examples

			1, 1, 5/2, 31/6, 265/24, 2621/120, 31621/720, 85319/1008, 6574961/40320, ...
		

Crossrefs

Cf. A000041, A017665, A058892, A066186, A067764, A098987, A380271 (denominators).

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[-1 + 1/Product[1 - x^k, {k, 1, nmax}]], {x, 0, nmax}], x] // Numerator
b[0] = 1; b[n_] := b[n] = (1/n) Sum[k PartitionsP[k] b[n - k], {k, 1, n}]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 20}]

Formula

b(0) = 1, b(n) = (1/n) * Sum_{k=1..n} k * A000041(k) * b(n-k), a(n) = numerator of b(n).