A360742 Number T(n,k) of sets of nonempty 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, 2, 0, 1, 3, 3, 2, 0, 1, 4, 6, 5, 3, 0, 1, 5, 10, 10, 7, 4, 0, 1, 6, 14, 19, 16, 10, 5, 0, 1, 7, 19, 30, 32, 24, 14, 6, 0, 1, 8, 26, 46, 57, 52, 35, 19, 8, 0, 1, 9, 32, 67, 94, 97, 79, 50, 25, 10, 0, 1, 10, 40, 93, 147, 172, 157, 117, 69, 33, 12

Offset: 0

Author

Alois P. Heinz, Feb 18 2023

Keywords

• nonn
• tabl

Examples

			T(6,3) = 10: {[1,1,4]}, {[1,2,3]}, {[2,2,2]}, {[1],[1,4]}, {[1],[2,3]}, {[2],[1,3]}, {[2],[2,2]}, {[3],[1,2]}, {[4],[1,1]}, {[1],[2],[3]}.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2,  2;
  0, 1, 3,  3,  2;
  0, 1, 4,  6,  5,  3;
  0, 1, 5, 10, 10,  7,  4;
  0, 1, 6, 14, 19, 16, 10,  5;
  0, 1, 7, 19, 30, 32, 24, 14,  6;
  0, 1, 8, 26, 46, 57, 52, 35, 19,  8;
  0, 1, 9, 32, 67, 94, 97, 79, 50, 25, 10;
  ...
		

Links

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

Crossrefs

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

Main diagonal gives A000009.

T(n+2,n+1) gives A036469.

Row sums give A261049.

T(2n,n) gives A360714.

Cf. A000041, A055884 (similar triangle for multisets), A330463.

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)))))
    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), 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, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..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]]]]];
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], {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_, k_] := Coefficient[b[n, n], x, k];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 15 2023, after Alois P. Heinz *)

Formula

T(n,n) + T(n+1,n) = T(n+2,n+1) for n>=0.