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A369697 Number of unordered pairs (p,q) of distinct partitions of n such that the set of parts in q is equal to the set of parts in p.

%I A369697 #21 Feb 29 2024 13:00:20
%S A369697 0,0,0,0,0,1,1,4,5,12,19,35,53,91,136,225,325,505,741,1107,1590,2340,
%T A369697 3313,4748,6682,9412,13091,18241,25080,34478,47118,64069,86698,117012,
%U A369697 157121,210189,280385,372309,493279,650905,856913,1123675,1471196,1918293,2497470
%N A369697 Number of unordered pairs (p,q) of distinct partitions of n such that the set of parts in q is equal to the set of parts in p.
%H A369697 Alois P. Heinz, <a href="/A369697/b369697.txt">Table of n, a(n) for n = 0..350</a>
%F A369697 a(n) = (A369695(n) - A000041(n))/2.
%e A369697 a(5) = 1: (221, 2111).
%e A369697 a(6) = 1: (2211, 21111).
%e A369697 a(7) = 4: (22111, 211111), (2221, 211111), (2221, 22111), (331, 31111).
%e A369697 a(8) = 5: (221111, 2111111), (22211, 2111111), (22211, 221111), (3221, 32111), (3311, 311111).
%e A369697 a(9) = 12: (2211111, 21111111), (222111, 21111111), (222111, 2211111), (22221, 21111111), (22221, 2211111), (22221, 222111), (32211, 321111), (33111, 3111111), (3321, 321111), (3321, 32211), (4221, 42111), (441, 411111).
%e A369697 a(10) = 19: (22111111, 211111111), (2221111, 211111111), (2221111, 22111111), (222211, 211111111), (222211, 22111111), (222211, 2221111), (322111, 3211111), (32221, 3211111), (32221, 322111), (331111, 31111111), (33211, 3211111), (33211, 322111), (33211, 32221), (3331, 31111111), (3331, 331111), (42211, 421111), (4411, 4111111), (442, 4222), (5221, 52111).
%p A369697 b:= proc(n, m, i) option remember; `if`(n=0,
%p A369697      `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
%p A369697       b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
%p A369697     end:
%p A369697 a:= n-> (b(n$3)-combinat[numbpart](n))/2:
%p A369697 seq(a(n), n=0..50);
%t A369697 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 A369697 a[n_] := (b[n, n, n] - PartitionsP[n])/2;
%t A369697 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y A369697 Cf. A000041, A297388, A369695, A369696.
%K A369697 nonn
%O A369697 0,8
%A A369697 _Alois P. Heinz_, Jan 29 2024