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

0,9

Links

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

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A257654 A167637 A343489 * A109754 A220074 A059259

Adjacent sequences: A360739 A360740 A360741 * A360743 A360744 A360745

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 18 2023

Status

approved