A360785 Number of multisets of nonempty strict integer partitions with a total of 2n parts and total sum of 3n.

1, 2, 5, 12, 26, 54, 112, 220, 427, 812, 1518, 2790, 5074, 9096, 16144, 28360, 49367, 85180, 145867, 247886, 418426, 701702, 1169673, 1938498, 3195497, 5240386, 8552308, 13892638, 22468406, 36184636, 58040397, 92737842, 147631545, 234184172, 370215442, 583343070

Offset: 0

Author

Alois P. Heinz, Feb 20 2023

Keywords

• nonn

Examples

			a(2) = 5: {[1],[1],[1],[3]}, {[1],[1],[2],[2]}, {[1],[1],[1,3]}, {[1],[2],[1,2]}, {[1,2],[1,2]}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A360763, A360784.

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(3*n$2), x, 2*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[3n, 3n], x, 2n];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Dec 09 2023, after Alois P. Heinz *)

Formula

a(n) = A360763(3n,2n) = A360763(3n+j,2n+j) for j>=0.

a(n) = max({ A360763(k,k-n) : k>=n }).