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A285230 Number of multisets of exactly n partitions of positive integers into distinct parts with total sum of parts equal to 2n.

%I A285230 #19 Dec 11 2020 13:39:45
%S A285230 1,1,3,5,11,19,37,63,115,195,339,566,957,1573,2599,4217,6842,10962,
%T A285230 17531,27767,43862,68769,107469,166942,258461,398124,611237,934356,
%U A285230 1423724,2161145,3270560,4932647,7418099,11121610,16629101,24794130,36874451,54698714
%N A285230 Number of multisets of exactly n partitions of positive integers into distinct parts with total sum of parts equal to 2n.
%H A285230 Alois P. Heinz, <a href="/A285230/b285230.txt">Table of n, a(n) for n = 0..10000</a>
%F A285230 G.f.: Product_{j>=1} 1/(1-x^j)^A000009(j+1).
%F A285230 a(n) = A285229(2n,n).
%e A285230 a(3) = 5: {4,1,1}, {31,1,1}, {3,2,1}, {21,2,1}, {2,2,2}.
%p A285230 with(numtheory):
%p A285230 g:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A285230      `if`(d::odd, d, 0), d=divisors(j))*g(n-j), j=1..n)/n)
%p A285230     end:
%p A285230 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A285230       d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A285230     end:
%p A285230 seq(a(n), n=0..50);
%t A285230 g[n_] := g[n] = If[n == 0, 1, Sum[Sum[If[OddQ[d], d, 0], {d, Divisors[j]}]* g[n - j], {j, 1, n}]/n];
%t A285230 a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d + 1], {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
%t A285230 a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 11 2020, after _Alois P. Heinz_ *)
%Y A285230 Cf. A000009, A285229.
%K A285230 nonn
%O A285230 0,3
%A A285230 _Alois P. Heinz_, Apr 14 2017