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

1,2

Comments

Stirling transform of A000005.

Links

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

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).

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}]

Crossrefs

Cf. A000005, A160399, A308555.

Sequence in context: A134168 A124427 A350589 * A055730 A120018 A354120

Adjacent sequences: A308551 A308552 A308553 * A308555 A308556 A308557

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 07 2019

Status

approved