A303970 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^H(k), where H(k) is the k-th harmonic number.

1, 1, 5, 26, 199, 1599, 17053, 186276, 2460057, 34226729, 537669401, 8925732958, 163894885735, 3151342927823, 65678713377873, 1437541042260704, 33545591623360881, 819213454875992337, 21170268780829522093, 570252657062810041954, 16139888268919495959911, 475126022355752304699455, 14608848314409377281498213

Offset

0,3

Comments

a(n)/n! is the Euler transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..436

N. J. A. Sloane, Transforms

Formula

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

Maple

H:= proc(n) option remember; `if`(n=0, 0, 1/n+H(n-1)) end:
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      H(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
    end:
a:= n-> n!*b(n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 03 2018

Mathematica

nmax = 22; CoefficientList[Series[Product[1/(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}]

Crossrefs

Cf. A001008, A002805, A028342.

Sequence in context: A175151 A121750 A143341 * A366215 A348206 A007286

Adjacent sequences: A303967 A303968 A303969 * A303971 A303972 A303973

Keyword

nonn

Author

Ilya Gutkovskiy, May 03 2018

Status

approved