A308554 - OEIS

%I #10 Aug 04 2021 09:40:34

%S 1,3,9,30,113,472,2145,10514,55428,313255,1886888,12029741,80701715,

%T 567541878,4175795147,32104799401,257561662496,2151841672173,

%U 18676002357864,167951667633495,1561420657033927,14980472336450530,148140814019762129,1508776236781766431

%N Expansion of e.g.f. Sum_{k>=1} tau(k)*(exp(x) - 1)^k/k!, where tau = number of divisors (A000005).

%C Stirling transform of A000005.

%H Alois P. Heinz, <a href="/A308554/b308554.txt">Table of n, a(n) for n = 1..575</a>

%F G.f.: Sum_{k>=1} tau(k)*x^k / Product_{j=1..k} (1 - j*x).

%F a(n) = Sum_{k=1..n} Stirling2(n,k)*tau(k).

%p b:= proc(n, m) option remember; uses numtheory;

%p      `if`(n=0, tau(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 04 2021

%t nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]

%t nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]

%t Table[Sum[StirlingS2[n, k] DivisorSigma[0, k], {k, 1, n}], {n, 1, 24}]

%Y Cf. A000005, A160399, A308555.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jun 07 2019