A305203 Expansion of e.g.f. Product_{k>=1} (1 + H(k)*x^k), where H(k) is the k-th harmonic number.

1, 1, 3, 20, 94, 854, 7638, 77678, 823184, 11711952, 162710640, 2405290392, 40661618688, 701353671264, 13592382983424, 280431464804640, 5835146351362560, 130171240155651840, 3168997587241864704, 77082927941097660672, 2037627154674197591040, 56017463733173686947840

Offset: 0

Author

Ilya Gutkovskiy, May 27 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..441

Crossrefs

Cf. A001008, A002805, A007838, A304494, A305201.

Programs

Maple

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 27 2018

Mathematica

nmax = 21; CoefficientList[Series[Product[(1 + HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) d HarmonicNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]

Formula

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

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