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

0,3

Links

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

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

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}]

Prog

(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

Crossrefs

Cf. A088009, A318769, A318696.

Cf. A000262, A305127, A318695.

Cf. A053529, A028342.

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

Sequence in context: A355396 A265906 A060389 * A203122 A207157 A198049

Adjacent sequences: A318914 A318915 A318916 * A318918 A318919 A318920

Keyword

nonn

Author

Vaclav Kotesovec, Sep 05 2018

Status

approved