A308345 Expansion of e.g.f. Sum_{k>=1} log(1/(1 - x^k/k)).

1, 2, 4, 15, 48, 310, 1440, 11970, 85120, 821016, 7257600, 91707000, 958003200, 13440913200, 178919989248, 2809456650000, 41845579776000, 763629026160000, 12804747411456000, 257140635922025856, 4918792391884800000, 106876408948152480000

Offset: 1

Author

Ilya Gutkovskiy, May 21 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..450

Crossrefs

Cf. A007841, A038048, A182926, A182927, A308337.

Programs

Mathematica

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

Formula

a(n) = n! * Sum_{d|n} 1/(d*(n/d)^d).

a(n) = A007841(n) - (1/n) * Sum_{k=1..n-1} k*binomial(n,k)*A007841(n-k)*a(k).

a(n) ~ 2 * (n-1)!. - Vaclav Kotesovec, Feb 16 2020