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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[n - i*j, m - i*h, i - 1], {h, 0, m/i}], {j, 1, n/i}]]];
a[n_] := b[n, n, n] - PartitionsP[n];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)
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Number of pairs (p,q) of distinct partitions of n with p != q such that the set of parts in q is a subset of the set of parts in p.
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Alois P. Heinz, <a href="/A369707/b369707_1.txt">Table of n, a(n) for n = 0..350</a>
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