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

%I A300511 #11 Mar 11 2018 17:37:22
%S A300511 1,1,3,10,42,203,1119,6841,45916,334414,2622256,21984668,195991611,
%T A300511 1849158088,18390563792,192128761836,2102097270199,24022460183508,
%U A300511 286060559298908,3542047217686560,45517563689858955,606014811356799054,8346153294214800894,118731713512110007282
%N A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).
%C A300511 Exponential transform of A000041.
%H A300511 Alois P. Heinz, <a href="/A300511/b300511.txt">Table of n, a(n) for n = 0..533</a>
%H A300511 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A300511 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A300511 E.g.f.: exp(Sum_{k>=1} A000041(k)*x^k/k!).
%e A300511 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! + ..
%p A300511 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A300511       binomial(n-1, j-1)*combinat[numbpart](j), j=1..n))
%p A300511     end:
%p A300511 seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 07 2018
%t A300511 nmax = 23; CoefficientList[Series[Exp[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300511 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}]
%Y A300511 Cf. A000041, A001970, A058892, A215915, A218481, A300512.
%K A300511 nonn
%O A300511 0,3
%A A300511 _Ilya Gutkovskiy_, Mar 07 2018