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A300632 Expansion of e.g.f. exp(x + Sum_{k>=2} prime(k-1)*x^k/k!).

%I A300632 #7 Mar 15 2018 12:30:58
%S A300632 1,1,3,10,42,203,1119,6839,45895,334142,2619052,21946647,195537777,
%T A300632 1843619725,18321431155,191242913022,2090436115146,23864653888881,
%U A300632 283865214366771,3510656353388517,45056394441558593,599057016471131604,8238406603745152620,117020080948487107289
%N A300632 Expansion of e.g.f. exp(x + Sum_{k>=2} prime(k-1)*x^k/k!).
%C A300632 Exponential transform of A008578.
%H A300632 Alois P. Heinz, <a href="/A300632/b300632.txt">Table of n, a(n) for n = 0..534</a>
%H A300632 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A300632 E.g.f.: exp(Sum_{k>=1} A008578(k)*x^k/k!).
%e A300632 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! + 6839*x^7/7! + ..
%p A300632 a:= proc(n) option remember; (p-> `if`(n=0, 1, add(a(n-j)*p(j)*
%p A300632       binomial(n-1, j-1), j=1..n)))(t-> `if`(t=1, 1, ithprime(t-1)))
%p A300632     end:
%p A300632 seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 10 2018
%t A300632 nmax = 23; CoefficientList[Series[Exp[x + Sum[Prime[k - 1] x^k/k!, {k, 2, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t A300632 p[1] = 1; p[n_] := p[n] = Prime[n - 1]; a[n_] := a[n] = Sum[p[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
%Y A300632 Cf. A000040, A007446, A008578, A023626, A030011, A030012, A030013, A030014, A030015, A030016.
%K A300632 nonn
%O A300632 0,3
%A A300632 _Ilya Gutkovskiy_, Mar 10 2018