A061198 Square table by antidiagonals where T(n,k) is number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 3, 4, 3, 2, 1, 1, 0, 4, 5, 4, 3, 2, 1, 1, 0, 5, 7, 6, 5, 3, 2, 1, 1, 0, 6, 9, 9, 6, 5, 3, 2, 1, 1, 0, 8, 13, 12, 10, 7, 5, 3, 2, 1, 1, 0, 10, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 0, 12, 22, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 15, 27, 29, 25, 20, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Author

Henry Bottomley, Apr 20 2001

Keywords

• nonn
• tabl

Examples

			Square table T(n,k) begins:
  1, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, ...
  1, 1, 1, 2, 2, 3,  4,  5,  6,  8, 10, ...
  1, 1, 2, 2, 4, 5,  7,  9, 13, 16, 22, ...
  1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, ...
  1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ...
  1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ...
  1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, ...
  1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...
		

Crossrefs

Rows include A000007, A000009, A000726, A035959.

Main diagonal is A000041.

A061199 is the same table but excluding cases where n>k.

Cf. A286653.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..13);  # Alois P. Heinz, Jan 26 2023

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k], {j, 0, Min[n/i, k]}]]];
A[n_, k_] := b[k, k, n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Feb 11 2023, after Alois P. Heinz *)

Formula

G.f. for row n of table: Product_{j>=1} Sum_{k=0..n} x^(j*k) = Product_{j>=1} (1-x^((n+1)*j)) / (1-x^j). - Sean A. Irvine, Jan 26 2023