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A319111
Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).
2
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Sep 10 2018
STATUS
approved