A293554 - OEIS

%I #8 Oct 17 2017 18:09:57

%S 1,1,3,10,45,216,1232,7624,52215,385495,3056680,25825669,231503636,

%T 2191866327,21835650219,228089127908,2490775088645,28362322146780,

%U 336015253520857,4133561828779865,52705520063966840,695406327616587268,9480212057583970983

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

%H Alois P. Heinz, <a href="/A293554/b293554.txt">Table of n, a(n) for n = 0..537</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^binomial(n+k-2,n-1).

%F a(n) = A293551(n,n).

%p with(numtheory):

%p b:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*

%p       binomial(d+k-2, k-1), d=divisors(j))*b(n-j, k), j=1..n)/n)

%p     end:

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

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 17 2017

%t Table[SeriesCoefficient[E^(Sum[x^k/(k (1 - x^k)^n), {k, 1, n}]), {x, 0, n}], {n, 0, 22}]

%t Table[SeriesCoefficient[Product[1/(1 - x^k)^Binomial[n + k - 2, n - 1], {k, 1, n}], {x, 0, n}], {n, 0, 22}]

%Y Main diagonal of A293551.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Oct 11 2017