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A320567 Expansion of e.g.f. exp(x) * Product_{k>=1} (1 + x^k/k!).

%I A320567 #9 Sep 27 2019 22:47:07
%S A320567 1,2,4,11,32,97,355,1423,5696,23141,108149,559693,2913971,14806365,
%T A320567 75692999,432849976,2780749376,18237870285,115493756737,708062095921,
%U A320567 4354275076517,29539724932771,227955214198529,1836106089485736,14279737884301139,105409744347318897
%N A320567 Expansion of e.g.f. exp(x) * Product_{k>=1} (1 + x^k/k!).
%C A320567 Binomial transform of A007837.
%H A320567 Alois P. Heinz, <a href="/A320567/b320567.txt">Table of n, a(n) for n = 0..692</a>
%H A320567 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A320567 E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)).
%F A320567 a(n) = Sum_{k=0..n} binomial(n,k)*A007837(k).
%p A320567 seq(coeff(series(factorial(n)*exp(x)*mul(1+x^k/factorial(k),k=1..n),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Oct 15 2018
%p A320567 # second Maple program:
%p A320567 b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
%p A320567        d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
%p A320567     end:
%p A320567 a:= n-> add(b(n-i)*binomial(n, i), i=0..n):
%p A320567 seq(a(n), n=0..27);  # _Alois P. Heinz_, Sep 27 2019
%t A320567 nmax = 25; CoefficientList[Series[Exp[x] Product[(1 + x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A320567 nmax = 25; CoefficientList[Series[Exp[x + Sum[Sum[(-1)^(k + 1) x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%Y A320567 Cf. A007837, A320566.
%K A320567 nonn
%O A320567 0,2
%A A320567 _Ilya Gutkovskiy_, Oct 15 2018