A305201 Expansion of e.g.f. Product_{k>=1} 1/(1 - H(k)*x^k), where H(k) is the k-th harmonic number.

1, 1, 5, 26, 208, 1644, 18728, 201466, 2809672, 39505800, 647509992, 10851033984, 210456343392, 4090234000800, 89123794754304, 2000019423403824, 48674645933985408, 1217362548455301504, 32913123947574009984, 910006995701419453440, 26898048642355515339264

Offset: 0

Author

Ilya Gutkovskiy, May 27 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..430
• Vaclav Kotesovec, Graph - the asymptotic ratio (5000 terms)

Crossrefs

Cf. A001008, A002805, A007841, A303970, A305203.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[HarmonicNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d HarmonicNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]

Formula

E.g.f.: Product_{k>=1} 1/(1 - (A001008(k)/A002805(k))*x^k).

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

a(n) ~ n! * c * (3/2)^(n/2 + 1) / (3 - sqrt(6)), where c = Product_{k>=3} 1/(1 - (2/3)^(k/2) * H(k)) = 20723937.5142714953478411012151498609843924051679047516... - Vaclav Kotesovec, Nov 05 2019