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%I #15 Sep 28 2019 18:04:46
%S 1,2,6,23,110,617,4035,29927,249926,2316317,23674841,264329177,
%T 3207278255,42011308653,591460307157,8905905152798,142897741683846,
%U 2433947385964373,43873382718719949,834402502632550589,16699964488044322205,350869837371828862607,7721899536993122262447
%N Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - x^k/k!).
%C Binomial transform of A005651.
%H Vaclav Kotesovec, <a href="/A320566/b320566.txt">Table of n, a(n) for n = 0..440</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).
%F a(n) = Sum_{k=0..n} binomial(n,k)*A005651(k).
%F a(n) ~ exp(1) * A247551 * n!. - _Vaclav Kotesovec_, Jul 21 2019
%p seq(coeff(series(factorial(n)*exp(x)*mul((1-x^k/factorial(k))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # _Muniru A Asiru_, Oct 15 2018
%t nmax = 22; CoefficientList[Series[Exp[x] Product[1/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 22; CoefficientList[Series[Exp[x + Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[Binomial[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]
%Y Cf. A005651, A320567, A327870.
%Y Row sums of A327801.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 15 2018
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