A308457 Expansion of e.g.f. (1/(1 - x)) * Product_{k>=2} 1/(1 - x^k)^(phi(k)/2), where phi() is the Euler totient function (A000010).

1, 1, 3, 15, 93, 765, 6615, 73395, 855225, 11348505, 163593675, 2633729175, 44537325525, 829112008725, 16299062754975, 340762189642875, 7597436750528625, 178862527106888625, 4426363064514265875, 115222810432347993375, 3139125774622690978125

Offset: 0

Author

Ilya Gutkovskiy, May 27 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..435

Crossrefs

Cf. A000010, A023022, A023896, A051193, A057661, A061255.

Programs

Mathematica

nmax = 20; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k)^(EulerPhi[k]/2), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[LCM[k, j], {j, 1, k}] x^k/k^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Total[Numerator[Range[k]/k]] k! Binomial[n - 1, k - 1] a[n - k]/k, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 20}]

Formula

E.g.f.: exp(Sum_{k>=1} A057661(k)*x^k/k).

E.g.f.: exp(Sum_{k>=1} A051193(k)*x^k/k^2).

E.g.f.: d/dx ( exp(arctanh(x)) ) * Product_{k>=3} 1/(1 - x^k)^A023022(k).

a(n) ~ A * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi)^(2/3) - n - 1/12) * n^(n + 1/36) / (2^(1/9) * 3^(19/36) * (Pi*Zeta(3))^(1/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 28 2019

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A023896(k)/k). - Ilya Gutkovskiy, May 28 2019