%I #18 Feb 29 2024 12:59:59
%S 1,1,2,3,5,8,12,19,27,42,61,91,130,192,271,401,556,802,1126,1597,2217,
%T 3132,4315,6003,8257,11370,15527,21251,28798,39043,52722,70911,95047,
%U 127155,169431,225072,298362,393946,519294,682090,894251,1168258,1524370,1981554
%N Number of unordered pairs (p,q) of partitions of n such that the set of parts in q is equal to the set of parts in p.
%H Alois P. Heinz, <a href="/A369696/b369696.txt">Table of n, a(n) for n = 0..350</a>
%F a(n) = (A000041(n) + A369695(n))/2.
%e a(5) = 8: (11111, 11111), (2111, 2111), (2111, 221), (221, 221), (311, 311), (32, 32), (41, 41), (5, 5).
%p b:= proc(n, m, i) option remember; `if`(n=0,
%p `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
%p b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
%p end:
%p a:= n-> (b(n$3)+combinat[numbpart](n))/2:
%p seq(a(n), n=0..50);
%t b[n_, m_, i_] := b[n, m, i] = If[n == 0, If[m == 0, 1, 0], If[i < 1, 0, b[n, m, i-1] + Sum[Sum[b[Sequence @@ Sort[{n-i*j, m-i*h}], i-1], {h, 1, m/i}], {j, 1, n/i}]]];
%t a[n_] := (b[n, n, n] + PartitionsP[n])/2;
%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y Cf. A000041, A297388, A369695, A369697.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jan 29 2024