A324236 Expansion of e.g.f. 1 / (1 - Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

1, 1, 4, 21, 149, 1317, 13985, 173209, 2451844, 39044784, 690862770, 13446615722, 285510978887, 6567419023617, 162686428939423, 4317885767971448, 122241788335870103, 3677030054440996775, 117111150680951037907, 3937135961534144480556, 139328182441566999124409

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..409

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * p(k) * a(n-k).

a(n) ~ c * d^n * n!, where d = 1.769410350604938716841596133605930996231892313627986058432895713767619380283... and c = 0.6329116440270047042622953043644713645679657251851049998748689226219... - Vaclav Kotesovec, Sep 03 2019

a(n) = Sum_{ordered partitions S of [n]} Product_{blocks B of S} A000041(|B|). a(3) = 21 = 1*3 + 6*(1*2) + 6*(1*1*1) from 123, 12|3, 3|12, 13|2, 2|13, 1|23, 23|1, 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1. - Md. Rad Sarar Anando, Jan 28 2026

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-k)*
      binomial(n, k)*combinat[numbpart](k), k=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 02 2019

Mathematica

nmax = 20; CoefficientList[Series[1/(1 - Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] PartitionsP[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Prog

(PARI) A324236(n)=my(A=vector(n+1)); A[1]=1; for(i=2, n+1, A[i]=sum(k=1, i-1, binomial(i-1, k)*numbpart(k)*A[i-k])); A[n+1] \\ Md. Rad Sarar Anando, Jan 29 2026

Crossrefs

Cf. A000041, A055887, A218481, A300511, A302199, A324237, A000670.

Sequence in context: A277505 A183387 A346430 * A392981 A330019 A025164

Adjacent sequences: A324233 A324234 A324235 * A324237 A324238 A324239

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 02 2019

Status

approved