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A293554 a(n) = [x^n] exp(Sum_{k>=1} x^k/(k*(1 - x^k)^n)).

%I A293554 #8 Oct 17 2017 18:09:57
%S A293554 1,1,3,10,45,216,1232,7624,52215,385495,3056680,25825669,231503636,
%T A293554 2191866327,21835650219,228089127908,2490775088645,28362322146780,
%U A293554 336015253520857,4133561828779865,52705520063966840,695406327616587268,9480212057583970983
%N A293554 a(n) = [x^n] exp(Sum_{k>=1} x^k/(k*(1 - x^k)^n)).
%H A293554 Alois P. Heinz, <a href="/A293554/b293554.txt">Table of n, a(n) for n = 0..537</a>
%H A293554 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A293554 a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^binomial(n+k-2,n-1).
%F A293554 a(n) = A293551(n,n).
%p A293554 with(numtheory):
%p A293554 b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
%p A293554       binomial(d+k-2, k-1), d=divisors(j))*b(n-j, k), j=1..n)/n)
%p A293554     end:
%p A293554 a:= n-> b(n$2):
%p A293554 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 17 2017
%t A293554 Table[SeriesCoefficient[E^(Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]), {x, 0, n}], {n, 0, 22}]
%t A293554 Table[SeriesCoefficient[Product[1/(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 22}]
%Y A293554 Main diagonal of A293551.
%K A293554 nonn
%O A293554 0,3
%A A293554 _Ilya Gutkovskiy_, Oct 11 2017