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

0,2

Links

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

Formula

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

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

Example

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

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 *)

Crossrefs

Cf. A360763, A360784.

Sequence in context: A262803 A026622 A297618 * A198896 A026688 A116726

Adjacent sequences: A360782 A360783 A360784 * A360786 A360787 A360788

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2023

Status

approved