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

1,2

Comments

Stirling transform of A000010.

Links

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

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).

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}]

Crossrefs

Cf. A000010, A306988.

Sequence in context: A053617 A089449 A264601 * A226435 A292318 A150271

Adjacent sequences: A308561 A308562 A308563 * A308565 A308566 A308567

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 07 2019

Status

approved