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

0,3

Links

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

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)

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}]

Crossrefs

Cf. A000010, A061255, A159929, A318917.

Cf. A279784, A306327, A316961.

Sequence in context: A101268 A188920 A281362 * A128768 A235607 A254007

Adjacent sequences: A319108 A319109 A319110 * A319112 A319113 A319114

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 10 2018

Status

approved