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

0,3

Links

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

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).

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<n, 0,
     `if`(n=0, 1, b(n, i-1)+H(i)*b(n-i, min(n-i, i-1))))
    end:
a:= n-> 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}]

Crossrefs

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

Sequence in context: A092786 A015529 A278319 * A246150 A000948 A074831

Adjacent sequences: A305200 A305201 A305202 * A305204 A305205 A305206

Keyword

nonn

Author

Ilya Gutkovskiy, May 27 2018

Status

approved