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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).

%I A324236 #18 Sep 03 2019 03:54:08
%S A324236 1,1,4,21,149,1317,13985,173209,2451844,39044784,690862770,
%T A324236 13446615722,285510978887,6567419023617,162686428939423,
%U A324236 4317885767971448,122241788335870103,3677030054440996775,117111150680951037907,3937135961534144480556,139328182441566999124409
%N 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).
%H A324236 Vaclav Kotesovec, <a href="/A324236/b324236.txt">Table of n, a(n) for n = 0..409</a>
%F A324236 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * p(k) * a(n-k).
%F A324236 a(n) ~ c * d^n * n!, where d = 1.769410350604938716841596133605930996231892313627986058432895713767619380283... and c = 0.6329116440270047042622953043644713645679657251851049998748689226219... - _Vaclav Kotesovec_, Sep 03 2019
%p A324236 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-k)*
%p A324236       binomial(n, k)*combinat[numbpart](k), k=1..n))
%p A324236     end:
%p A324236 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 02 2019
%t A324236 nmax = 20; CoefficientList[Series[1/(1 - Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%t A324236 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 A324236 Cf. A000041, A055887, A218481, A300511, A302199, A324237.
%K A324236 nonn
%O A324236 0,3
%A A324236 _Ilya Gutkovskiy_, Sep 02 2019