A305201 - OEIS

%I #13 Dec 04 2024 16:10:04

%S 1,1,5,26,208,1644,18728,201466,2809672,39505800,647509992,

%T 10851033984,210456343392,4090234000800,89123794754304,

%U 2000019423403824,48674645933985408,1217362548455301504,32913123947574009984,910006995701419453440,26898048642355515339264

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

%H Vaclav Kotesovec, <a href="/A305201/b305201.txt">Table of n, a(n) for n = 0..430</a>

%H Vaclav Kotesovec, <a href="/A305201/a305201.jpg">Graph - the asymptotic ratio (5000 terms)</a>

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

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

%F 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

%t nmax = 20; CoefficientList[Series[Product[1/(1 - HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t 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]!

%t 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}]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 27 2018