A336031 Number of compositions of n such that the set of parts and the set of multiplicities of parts are equal.

1, 1, 0, 0, 4, 3, 4, 0, 5, 1, 70, 120, 122, 130, 446, 0, 277, 726, 370, 1064, 13751, 38913, 41272, 81168, 137014, 84448, 300642, 490540, 517806, 341033, 1467180, 425328, 2403512, 2916863, 4455856, 39855808, 164203236, 216675811, 447273890, 730795760, 1154455982

Offset: 0

Author

Alois P. Heinz, Jul 07 2020

Keywords

• nonn

Crossrefs

Cf. A114640, A336032.

Programs

Maple

b:= proc(n, i, p, f, g) option remember; `if`(n=0, `if`(f=g, p!, 0),
      `if`(i<1, 0, add(b(n-i*j, i-1, p+j, `if`(j=0, f, {f[], i}),
      `if`(j=0, g, {g[], j}))/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0, {}$2):
seq(a(n), n=0..32);

Mathematica

b[n_, i_, p_, f_, g_] := b[n, i, p, f, g] = If[n == 0, If[f == g, p!, 0],
     If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j,
     If[j == 0, f, Union@Append[f, i]],
     If[j == 0, g, Union@Append[g, j]]]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0, {}, {}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 13 2022, after Alois P. Heinz *)