A304494 -id:A304494 - cp's OEIS frontend

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

Ilya Gutkovskiy, May 27 2018

nonn

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

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

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

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}]

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).

A304496 Expansion of e.g.f. Product_{k>=1} (1 - x^k)^H(k), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, -1, -3, -2, 3, 261, 745, 12412, 16289, -260081, -5424199, -96985734, -2047127621, -17402659299, -84365982987, -2937186832544, 39650368238977, 1047895936025183, 35975009604881845, 638531451763185398, 14668256344792565331, 248159858571597211093, 6320237684944085611809

Offset: 0

Ilya Gutkovskiy, May 13 2018

sign

N. J. A. Sloane, Transforms

Cf. A001008, A002805, A028343, A303970, A304494.

nmax = 22; CoefficientList[Series[Product[(1 - x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d HarmonicNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]

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

cp's OEIS Frontend

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

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