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

0,9

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.

Links

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

Formula

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

Example

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;
  ...

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 *)

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.

Sequence in context: A321391 A244003 A369738 * A332670 A118344 A343138

Adjacent sequences: A360760 A360761 A360762 * A360764 A360765 A360766

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 19 2023

Status

approved