A306023 Stirling transform of partitions into distinct parts (A000009).

1, 1, 2, 6, 22, 89, 391, 1875, 9822, 55817, 340535, 2208681, 15118109, 108677575, 817914056, 6431115486, 52741729600, 450432487463, 3999401133601, 36853795902353, 351799243932131, 3472526583025397, 35382850151528847, 371592232539942447, 4016792440158613798

Offset: 0

Author

Vaclav Kotesovec, Jun 17 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..573
• Eric Weisstein's World of Mathematics, Stirling Transform.

Crossrefs

Cf. A000009, A305550, A306022.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(b(j)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 17 2018

Mathematica

Table[Sum[StirlingS2[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 25}]

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k)*A000009(k).