A288267 Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

1, 1, 1, 1, 1, 2, 2, 1, 3, 3, 1, 5, 5, 2, 9, 9, 4, 1, 15, 15, 6, 1, 26, 26, 11, 2, 45, 45, 19, 4, 78, 78, 33, 7, 1, 135, 135, 57, 12, 1, 234, 234, 99, 21, 2, 406, 406, 172, 37, 4, 704, 704, 298, 64, 7, 1222, 1222, 518, 112, 13, 1, 2120, 2120, 898, 194, 22, 1, 3679

Offset: 0

Author

Seiichi Manyama, Sep 01 2017

Keywords

• nonn
• tabf

Examples

			First few rows are:
   1;
   1,  1;
   1,  1;
   2,  2, 1;
   3,  3, 1;
   5,  5, 2;
   9,  9, 4, 1;
  15, 15, 6, 1.
		

Links

• Seiichi Manyama, Rows n = 0..481, flattened

Crossrefs

Columns 0+1,2 give A005169, A289080 (for n>0).

Cf. A003056.

Programs

Maple

T:= proc(n,k) option remember; `if`(k<0 or k*(k+1)/2>n, 0,
      `if`(n=0, 1, T(n, k+1)+T(n-k, k-1)))
    end:
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20); # Alois P. Heinz, Sep 01 2017

Mathematica

T[n_, k_] := T[n, k] = If[k < 0 || k(k+1)/2 > n, 0, If[n == 0, 1, T[n, k+1] + T[n-k, k-1]]];
Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1]-1)/2]}] // Flatten (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz *)