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
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..8200
Programs
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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
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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)