A318917 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k/k), where phi is the Euler totient function A000010.

1, 1, 2, 8, 38, 262, 1732, 16144, 153596, 1660796, 19415384, 264084064, 3664187848, 57366995272, 936097392752, 16131362629568, 302946516251408, 6034409270818576, 125044362929875744, 2756094464546395264, 63280996793936902496

Offset: 0

Author

Vaclav Kotesovec, Sep 05 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A088009, A318769, A318696.

Cf. A000262, A305127, A318695.

Cf. A053529, A028342.

Cf. A000010 (phi), A008683 (mu).

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[EulerPhi[k]* a[n-k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

PARI

a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 29 2022

Formula

a(n)/n! ~ 3^(1/4) * exp(2*sqrt(6*n)/Pi) / (Pi * 2^(3/4) * n^(3/4)).

E.g.f.: Product_{k>=1} 1 / (1 - x^k)^f(k), where f(k) = (1/k) * Sum_{j=1..k} mu(gcd(k,j)). - Ilya Gutkovskiy, Aug 17 2021

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 29 2022

E.g.f.: exp( Sum_{n>=1} (mu(n)/n) * x^n/(1 - x^n) ), where mu(n) = A008683(n). - Paul D. Hanna, Jun 24 2023