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

1, 3, 9, 30, 113, 472, 2145, 10514, 55428, 313255, 1886888, 12029741, 80701715, 567541878, 4175795147, 32104799401, 257561662496, 2151841672173, 18676002357864, 167951667633495, 1561420657033927, 14980472336450530, 148140814019762129, 1508776236781766431

Offset: 1

Author

Ilya Gutkovskiy, Jun 07 2019

Keywords

• nonn

Comments

Stirling transform of A000005.

Links

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

Crossrefs

Cf. A000005, A160399, A308555.

Programs

Maple

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, tau(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 04 2021

Mathematica

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

Formula

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

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