login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Expansion of e.g.f. Sum_{k>=1} sigma(k)*(exp(x) - 1)^k/k!, where sigma = sum of divisors (A000203).
3

%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