A308555 Expansion of e.g.f. Sum_{k>=1} sigma(k)*(exp(x) - 1)^k/k!, where sigma = sum of divisors (A000203).

1, 4, 14, 53, 222, 1011, 4944, 25884, 144963, 865556, 5477661, 36518635, 255323564, 1867122987, 14259709474, 113593734317, 942317654779, 8123227487723, 72599829900774, 671199117610868, 6407156027307909, 63061416571124056, 639303956718643041, 6670690645674913424

Offset: 1

Author

Ilya Gutkovskiy, Jun 07 2019

Keywords

• nonn

Comments

Stirling transform of A000203.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..575

Crossrefs

Cf. A000203, A185003, A308554.

Programs

Maple

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, sigma(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Aug 03 2021

Mathematica

nmax = 24; Rest[CoefficientList[Series[Sum[DivisorSigma[1, k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
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]]
Table[Sum[StirlingS2[n, k] DivisorSigma[1, k], {k, 1, n}], {n, 1, 24}]

Formula

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

a(n) = Sum_{k=1..n} Stirling2(n,k)*sigma(k).