A328193 Expansion of e.g.f. Sum_{k>=1} log(1/(1 + (-x)^k/k)).

1, 0, 4, 3, 48, 10, 1440, 1890, 85120, 49896, 7257600, 6883800, 958003200, 792277200, 178919989248, 194107914000, 41845579776000, 29714949264000, 12804747411456000, 12900082757417856, 4918792391884800000, 4594737608304480000, 2248001455555215360000

Offset: 1

Author

Ilya Gutkovskiy, Oct 30 2019

Keywords

• nonn

Links

• Georg Fischer, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A007838, A132960, A182927, A292358, A308338, A308345.

Programs

Maple

a:= n-> n!*add((-1)^(n-d)/(d*(n/d)^d), d=numtheory[divisors](n)):
seq(a(n), n=1..24);  # Alois P. Heinz, Oct 30 2019

Mathematica

nmax = 23; CoefficientList[Series[Sum[Log[1/(1 + (-x)^k/k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[n! Sum[(-1)^(n - d)/(d (n/d)^d), {d, Divisors[n]}], {n, 1, 23}]

Formula

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