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

0,3

Links

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

Formula

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

Example

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]}.

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 *)

Crossrefs

Cf. A000009, A360763, A360785.

Sequence in context: A036385 A196534 A317188 * A335464 A369733 A026635

Adjacent sequences: A360781 A360782 A360783 * A360785 A360786 A360787

Keyword

nonn

Author

Alois P. Heinz, Feb 20 2023

Status

approved