A318811 Expansion of e.g.f. exp(Sum_{k>=1} phi(k)*x^k), where phi is the Euler totient function A000010.

1, 1, 3, 19, 121, 1161, 9931, 124363, 1542129, 21594961, 335083411, 5712781251, 104044684393, 2036445474649, 42781075481691, 943820382272251, 22433542236603361, 556276331238284193, 14612462927067954979, 401110580118493111411, 11553483337639043003481

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..435

Eric Weisstein's World of Mathematics, Totient Function.

Wikipedia, Euler's totient function.

Formula

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

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * phi(k) * a(n-k)/(n-k)!. - Seiichi Manyama, Apr 07 2022

Mathematica

nmax = 25; CoefficientList[Series[Exp[Sum[EulerPhi[k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]!

Prog

(PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, eulerphi(k)*x^k)))) \\ Seiichi Manyama, Apr 07 2022
(PARI) a(n) = if(n==0, 1, (n-1)!*sum(k=1, n, k*eulerphi(k)*a(n-k)/(n-k)!)); \\ Seiichi Manyama, Apr 07 2022

Crossrefs

Cf. A061255, A061256, A006171.

Cf. A299069, A192065, A107742.

Cf. A294361, A294363.

Sequence in context: A294253 A293840 A327674 * A352299 A074572 A157455

Adjacent sequences: A318808 A318809 A318810 * A318812 A318813 A318814

Keyword

nonn

Author

Vaclav Kotesovec, Sep 04 2018

Status

approved