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.

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

Alois P. Heinz, Table of n, a(n) for n = 0..350

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

Cf. A000041, A040051, A297388, A369696, A369697.

Sequence in context: A018138 A336631 A228129 * A175125 A171925 A186946

Adjacent sequences: A369692 A369693 A369694 * A369696 A369697 A369698

Keyword

nonn

Author

Alois P. Heinz, Jan 29 2024

Status

approved