A308564 Expansion of e.g.f. Sum_{k>=1} phi(k)*(exp(x) - 1)^k/k!, where phi = Euler totient function (A000010).

1, 2, 6, 22, 90, 404, 1974, 10366, 57864, 341690, 2134022, 14104624, 98498972, 723664482, 5561589508, 44473028634, 368602225688, 3159852790392, 27997141025686, 256410638073082, 2428063270357748, 23774001479212114, 240580239864321604, 2513553050765310236

Offset: 1

Author

Ilya Gutkovskiy, Jun 07 2019

Keywords

• nonn

Comments

Stirling transform of A000010.

Links

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

Crossrefs

Cf. A000010, A306988.

Programs

Maple

b:= proc(n, m) option remember; uses numtheory;
     `if`(n=0, phi(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[EulerPhi[k] (Exp[x] - 1)^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
nmax = 24; Rest[CoefficientList[Series[Sum[EulerPhi[k] x^k/Product[(1 - j x), {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[StirlingS2[n, k] EulerPhi[k], {k, 1, n}], {n, 1, 24}]

Formula

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

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