A305203 - OEIS

%I #10 Jun 08 2018 23:39:09

%S 1,1,3,20,94,854,7638,77678,823184,11711952,162710640,2405290392,

%T 40661618688,701353671264,13592382983424,280431464804640,

%U 5835146351362560,130171240155651840,3168997587241864704,77082927941097660672,2037627154674197591040,56017463733173686947840

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

%H Alois P. Heinz, <a href="/A305203/b305203.txt">Table of n, a(n) for n = 0..441</a>

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

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

%p H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:

%p b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,

%p      `if`(n=0, 1, b(n, i-1)+H(i)*b(n-i, min(n-i, i-1))))

%p     end:

%p a:= n-> b(n$2)*n!:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, May 27 2018

%t nmax = 21; CoefficientList[Series[Product[(1 + HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!

%t 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]!

%t 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}]

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, May 27 2018