A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 5, 3, 1, 1, 0, 7, 3, 2, 1, 1, 0, 11, 6, 4, 2, 1, 1, 0, 15, 7, 5, 3, 2, 1, 1, 0, 22, 12, 7, 6, 3, 2, 1, 1, 0, 30, 14, 11, 7, 5, 3, 2, 1, 1, 0, 42, 22, 14, 11, 8, 5, 3, 2, 1, 1, 0, 56, 27, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 77, 40, 27, 21, 15, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Author

Seiichi Manyama, May 14 2023

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  1,  1;
  0,  5,  3,  1, 1;
  0,  7,  3,  2, 1, 1;
  0, 11,  6,  4, 2, 1, 1;
  0, 15,  7,  5, 3, 2, 1, 1;
  0, 22, 12,  7, 6, 3, 2, 1, 1;
  0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
  ...
		

Links

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

Crossrefs

Row sums give A323433.

Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.

T(2n,n) gives A052810.

Cf. A008284, A072233, A350890.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
    (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 14 2023

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

PARI

T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));

Formula

For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).