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%I #10 Aug 04 2021 09:35:15
%S 1,4,14,53,222,1011,4944,25884,144963,865556,5477661,36518635,
%T 255323564,1867122987,14259709474,113593734317,942317654779,
%U 8123227487723,72599829900774,671199117610868,6407156027307909,63061416571124056,639303956718643041,6670690645674913424
%N Expansion of e.g.f. Sum_{k>=1} sigma(k)*(exp(x) - 1)^k/k!, where sigma = sum of divisors (A000203).
%C Stirling transform of A000203.
%H Alois P. Heinz, <a href="/A308555/b308555.txt">Table of n, a(n) for n = 1..575</a>
%F G.f.: Sum_{k>=1} sigma(k)*x^k / Product_{j=1..k} (1 - j*x).
%F a(n) = Sum_{k=1..n} Stirling2(n,k)*sigma(k).
%p b:= proc(n, m) option remember; uses numtheory;
%p `if`(n=0, sigma(m), m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=1..24); # _Alois P. Heinz_, Aug 03 2021
%t nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[1, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
%t nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[Sum[StirlingS2[n, k] DivisorSigma[1, k], {k, 1, n}], {n, 1, 24}]
%Y Cf. A000203, A185003, A308554.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Jun 07 2019
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