A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the 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, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36

Offset: 0

Author

Henry Bottomley, Apr 20 2001

Keywords

• nonn
• tabl

Examples

			T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
   1, 1, 2, 2, 3,  4,  5,  6,  8, ...
      2, 2, 4, 5,  7,  9, 13, 16, ...
         3, 4, 6,  9, 12, 16, 22, ...
            5, 6, 10, 13, 19, 25, ...
               7, 10, 14, 20, 27, ...
                  11, 14, 21, 28, ...
                      15, 21, 29, ...
                          22, 29, ...
                              30, ...
		

Links

• Alois P. Heinz, Columns k = 0..140, flattened

Crossrefs

Rows effectively include A000007, A000009, A000726, A001935, A035959.

Main diagonal is A000041.

A061198 is the same table but includes cases where n>k.

T(n,2*n) gives: A232623.

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:
T:= (n, k)-> b(k$2, n):
seq(seq(T(n, k), n=0..k), k=0..12);  # Alois P. Heinz, Nov 27 2013

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]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)