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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.
3
1, 1, 2, 3, 5, 9, 13, 23, 32, 54, 80, 126, 183, 283, 407, 626, 881, 1307, 1867, 2704, 3807, 5472, 7628, 10751, 14939, 20782, 28618, 39492, 53878, 73521, 99840, 134980, 181745, 244167, 326552, 435261, 578747, 766255, 1012573, 1332995, 1751164, 2291933, 2995566
OFFSET
0,3
LINKS
FORMULA
a(n) = 2*A369696(n) - A000041(n).
a(n) = 2*A369697(n) + A000041(n).
a(n) mod 2 = A040051(n).
EXAMPLE
a(5) = 9: (11111, 11111), (2111, 2111), (2111, 221), (221, 2111), (221, 221), (311, 311), (32, 32), (41, 41), (5, 5).
MAPLE
b:= proc(n, m, i) option remember; `if`(n=0,
`if`(m=0, 1, 0), `if`(i<1, 0, b(n, m, i-1)+add(add(
b(sort([n-i*j, m-i*h])[], i-1), h=1..m/i), j=1..n/i)))
end:
a:= n-> b(n$3):
seq(a(n), n=0..50);
MATHEMATICA
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}]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 29 2024, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 29 2024
STATUS
approved