A330020 Expansion of e.g.f. Sum_{k>=1} x^k / (k! * (1 - x^k)^k).

1, 3, 7, 49, 121, 2161, 5041, 127681, 725761, 12852001, 39916801, 2917918081, 6227020801, 392423391361, 4740319584001, 122053759027201, 355687428096001, 57808258040332801, 121645100408832001, 18854997267794688001, 289799177540640768001, 7306005040298918553601

Offset: 1

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

• nonn

Crossrefs

Cf. A057625, A157019, A327578, A327579.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := n! Sum[(d + n/d - 2)!/(d! (d - 1)! (n/d - 1)!), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]

Formula

a(n) = n! * Sum_{d|n} (d + n/d - 2)! / (d! * (d - 1)! * (n/d - 1)!).