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Expansion of e.g.f. 1 / (1 - Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
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%I #18 Sep 03 2019 03:54:08

%S 1,1,4,21,149,1317,13985,173209,2451844,39044784,690862770,

%T 13446615722,285510978887,6567419023617,162686428939423,

%U 4317885767971448,122241788335870103,3677030054440996775,117111150680951037907,3937135961534144480556,139328182441566999124409

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

%H Vaclav Kotesovec, <a href="/A324236/b324236.txt">Table of n, a(n) for n = 0..409</a>

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

%F a(n) ~ c * d^n * n!, where d = 1.769410350604938716841596133605930996231892313627986058432895713767619380283... and c = 0.6329116440270047042622953043644713645679657251851049998748689226219... - _Vaclav Kotesovec_, Sep 03 2019

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-k)*

%p binomial(n, k)*combinat[numbpart](k), k=1..n))

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 02 2019

%t nmax = 20; CoefficientList[Series[1/(1 - Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

%t 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}]

%Y Cf. A000041, A055887, A218481, A300511, A302199, A324237.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 02 2019