cp's OEIS Frontend

A308554 Expansion of e.g.f. Sum_{k>=1} tau(k)*(exp(x) - 1)^k/k!, where tau = number of divisors (A000005).

%I A308554 #10 Aug 04 2021 09:40:34
%S A308554 1,3,9,30,113,472,2145,10514,55428,313255,1886888,12029741,80701715,
%T A308554 567541878,4175795147,32104799401,257561662496,2151841672173,
%U A308554 18676002357864,167951667633495,1561420657033927,14980472336450530,148140814019762129,1508776236781766431
%N A308554 Expansion of e.g.f. Sum_{k>=1} tau(k)*(exp(x) - 1)^k/k!, where tau = number of divisors (A000005).
%C A308554 Stirling transform of A000005.
%H A308554 Alois P. Heinz, <a href="/A308554/b308554.txt">Table of n, a(n) for n = 1..575</a>
%F A308554 G.f.: Sum_{k>=1} tau(k)*x^k / Product_{j=1..k} (1 - j*x).
%F A308554 a(n) = Sum_{k=1..n} Stirling2(n,k)*tau(k).
%p A308554 b:= proc(n, m) option remember; uses numtheory;
%p A308554      `if`(n=0, tau(m), m*b(n-1, m)+b(n-1, m+1))
%p A308554     end:
%p A308554 a:= n-> b(n, 0):
%p A308554 seq(a(n), n=1..24);  # _Alois P. Heinz_, Aug 04 2021
%t A308554 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 A308554 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 A308554 Table[Sum[StirlingS2[n, k] DivisorSigma[0, k], {k, 1, n}], {n, 1, 24}]
%Y A308554 Cf. A000005, A160399, A308555.
%K A308554 nonn
%O A308554 1,2
%A A308554 _Ilya Gutkovskiy_, Jun 07 2019