A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

1, 1, 3, 10, 42, 203, 1119, 6841, 45916, 334414, 2622256, 21984668, 195991611, 1849158088, 18390563792, 192128761836, 2102097270199, 24022460183508, 286060559298908, 3542047217686560, 45517563689858955, 606014811356799054, 8346153294214800894, 118731713512110007282

Offset

0,3

Comments

Exponential transform of A000041.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..533

N. J. A. Sloane, Transforms

Index entries for related partition-counting sequences

Formula

E.g.f.: exp(Sum_{k>=1} A000041(k)*x^k/k!).

Example

E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 42*x^4/4! + 203*x^5/5! + 1119*x^6/6! + 6841*x^7/7! + 45916*x^8/8! + ..

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*combinat[numbpart](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 07 2018

Mathematica

nmax = 23; CoefficientList[Series[Exp[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsP[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

Crossrefs

Cf. A000041, A001970, A058892, A215915, A218481, A300512.

Sequence in context: A091843 A361955 A300632 * A007552 A125274 A030903

Adjacent sequences: A300508 A300509 A300510 * A300512 A300513 A300514

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 07 2018

Status

approved