A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 174, 278, 447, 707, 1122, 1766, 2729, 4213, 6482, 9880, 15069, 22799, 34290, 51378, 76777, 114365, 169324, 250162, 368505, 540575, 792042, 1154798, 1680385, 2439101, 3530308, 5103380, 7349875, 10564955, 15155752, 21696072, 31007949, 44199845

Offset: 0

Author

Ilya Gutkovskiy, Sep 10 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..8200

Crossrefs

Cf. A000010, A061255, A159929, A318917.

Cf. A279784, A306327, A316961.

Programs

Maple

with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

Formula

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

From Vaclav Kotesovec, Feb 08 2019: (Start)

a(n) ~ c * 2^(2*n/5), where

c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0

c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1

c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2

c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3

c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4

(End)