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A369695 Number of ordered pairs (p,q) of partitions of n such that the set of parts in q is equal to the set of parts in p.

%I A369695 #18 Mar 02 2024 13:36:26
%S A369695 1,1,2,3,5,9,13,23,32,54,80,126,183,283,407,626,881,1307,1867,2704,
%T A369695 3807,5472,7628,10751,14939,20782,28618,39492,53878,73521,99840,
%U A369695 134980,181745,244167,326552,435261,578747,766255,1012573,1332995,1751164,2291933,2995566
%N A369695 Number of ordered 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 A369695 Alois P. Heinz, <a href="/A369695/b369695.txt">Table of n, a(n) for n = 0..350</a>
%F A369695 a(n) = 2*A369696(n) - A000041(n).
%F A369695 a(n) = 2*A369697(n) + A000041(n).
%F A369695 a(n) mod 2 = A040051(n).
%e A369695 a(5) = 9: (11111, 11111), (2111, 2111), (2111, 221), (221, 2111), (221, 221), (311, 311), (32, 32), (41, 41), (5, 5).
%p A369695 b:= proc(n, m, i) option remember; `if`(n=0,
%p A369695      `if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
%p A369695       b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
%p A369695     end:
%p A369695 a:= n-> b(n$3):
%p A369695 seq(a(n), n=0..50);
%t A369695 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 A369695 a[n_] := b[n, n, n];
%t A369695 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 29 2024, after _Alois P. Heinz_ *)
%Y A369695 Cf. A000041, A040051, A297388, A369696, A369697.
%K A369695 nonn
%O A369695 0,3
%A A369695 _Alois P. Heinz_, Jan 29 2024