A360784 Number of multisets of nonempty strict integer partitions with a total of n parts and total sum of 2n.

1, 1, 3, 8, 18, 39, 86, 175, 352, 688, 1318, 2472, 4576, 8322, 14959, 26560, 46657, 81130, 139866, 239047, 405496, 682891, 1142466, 1899344, 3139432, 5160455, 8438871, 13732292, 22242647, 35867937, 57597730, 92121145, 146775205, 232998683, 368579188, 581091003

Offset: 0

Author

Alois P. Heinz, Feb 20 2023

Keywords

• nonn

Examples

			a(3) = 8: {[1,2,3]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[3],[1,2]}, {[1],[1],[4]}, {[1],[2],[3]}, {[2],[2],[2]}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000009, A360763, A360785.

Programs

Maple

h:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, h(n, i-1)+x*h(n-i, min(n-i, i-1)))))
    end:
g:= proc(n, i, j) option remember; expand(`if`(j=0, 1, `if`(i<0, 0, add(
      g(n, i-1, j-k)*x^(i*k)*binomial(coeff(h(n$2), x, i)+k-1, k), k=0..j))))
    end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i<1, 0, add(b(n-i*j, i-1)*g(i$2, j), j=0..n/i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..35);

Mathematica

h[n_, i_] := h[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, h[n, i - 1] + x*h[n - i, Min[n - i, i - 1]]]]];
g[n_, i_, j_] := g[n, i, j] = Expand[If[j == 0, 1, If[i < 0, 0, Sum[g[n, i - 1, j - k]*x^(i*k)*Binomial[Coefficient[h[n, n], x, i] + k - 1, k], {k, 0, j}]]]];
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1]*g[i, i, j], {j, 0, n/i}]]]];
a[n_] := Coefficient[b[2 n, 2 n], x, n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 21 2023, after Alois P. Heinz *)

Formula

a(n) = A360763(2n,n).