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

%I A303970 #8 Sep 08 2018 05:22:41
%S A303970 1,1,5,26,199,1599,17053,186276,2460057,34226729,537669401,8925732958,
%T A303970 163894885735,3151342927823,65678713377873,1437541042260704,
%U A303970 33545591623360881,819213454875992337,21170268780829522093,570252657062810041954,16139888268919495959911,475126022355752304699455,14608848314409377281498213
%N A303970 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^H(k), where H(k) is the k-th harmonic number.
%C A303970 a(n)/n! is the Euler transform of [1, 1 + 1/2, 1 + 1/2 + 1/3, 1 + 1/2 + 1/3 + 1/4, ...].
%H A303970 Vaclav Kotesovec, <a href="/A303970/b303970.txt">Table of n, a(n) for n = 0..436</a>
%H A303970 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A303970 E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A001008(k)/A002805(k)).
%p A303970 H:= proc(n) option remember; `if`(n=0, 0, 1/n+H(n-1)) end:
%p A303970 b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
%p A303970       H(d), d=numtheory[divisors](j))*b(n-j), j=1..n)/n)
%p A303970     end:
%p A303970 a:= n-> n!*b(n):
%p A303970 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 03 2018
%t A303970 nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
%t A303970 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}]
%Y A303970 Cf. A001008, A002805, A028342.
%K A303970 nonn
%O A303970 0,3
%A A303970 _Ilya Gutkovskiy_, May 03 2018