A360763 Number T(n,k) of multisets of nonempty strict integer partitions with a total of k parts and total sum of n; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 4, 2, 1, 0, 1, 5, 8, 5, 2, 1, 0, 1, 6, 11, 10, 5, 2, 1, 0, 1, 7, 16, 18, 11, 5, 2, 1, 0, 1, 8, 22, 28, 22, 12, 5, 2, 1, 0, 1, 9, 28, 45, 39, 24, 12, 5, 2, 1, 0, 1, 10, 35, 63, 67, 46, 25, 12, 5, 2, 1, 0, 1, 11, 44, 89, 106, 86, 50, 26, 12, 5, 2, 1

Offset: 0

Author

Alois P. Heinz, Feb 19 2023

Keywords

• nonn
• tabl

Comments

T(n,k) is defined for all n >= 0 and k >= 0. Terms that are not in the triangle are zero.

Reversed rows and also the columns converge to A360785.

Examples

			T(6,1) = 1: {[6]}.
T(6,2) = 5: {[1],[5]}, {[2],[4]}, {[3],[3]}, {[1,5]}, {[2,4]}.
T(6,3) = 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]}.
T(6,4) = 5: {[1],[1],[1],[3]}, {[1],[1],[2],[2]}, {[1],[1],[1,3]}, {[1],[2],[1,2]}, {[1,2],[1,2]}.
T(6,5) = 2: {[1],[1],[1],[1],[2]}, {[1],[1],[1],[1,2]}.
T(6,6) = 1: {[1],[1],[1],[1],[1],[1]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2,  1;
  0, 1, 3,  2,  1;
  0, 1, 4,  4,  2,  1;
  0, 1, 5,  8,  5,  2,  1;
  0, 1, 6, 11, 10,  5,  2,  1;
  0, 1, 7, 16, 18, 11,  5,  2, 1;
  0, 1, 8, 22, 28, 22, 12,  5, 2, 1;
  0, 1, 9, 28, 45, 39, 24, 12, 5, 2, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Columns k=0-2 give: A000007, A057427, A001477(n-1) for n>=1.

Row sums give A089259.

T(2n,n) gives A360784.

T(3n,2n) gives A360785.

Cf. A000009, A008289, A055884, A285229, A360742, A360764.

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:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);

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}]]]];
T[n_] := CoefficientList[b[n, n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 12 2023, after Alois P. Heinz *)

Formula

T(3n,2n) = A360785(n) = T(3n+j,2n+j) for j>=0.