A137680 Triangle read by rows, T(n,k) = T(n-1, k-1) - T(n-k, k-1); with leftmost term in each row = sum of all previous terms.

1, 1, 1, 3, 0, 1, 7, 2, 0, 1, 17, 4, 1, 0, 1, 40, 10, 4, 1, 0, 1, 96, 23, 8, 3, 1, 0, 1, 228, 56, 19, 8, 3, 1, 0, 1, 544, 132, 46, 18, 7, 3, 1, 0, 1, 1296, 316, 109, 42, 18, 7, 3, 1, 0, 1, 3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1, 7361, 1793, 620, 241, 98, 41, 17, 7, 3, 1, 0, 1, 17544, 4272, 1477, 574, 233, 97, 40, 17, 7, 3, 1, 0, 1

Offset: 1

Author

Gary W. Adamson, Feb 05 2008

Keywords

• nonn
• tabl

Comments

A variation of the same sequence = column 2 of the triangle: (1, 0, 2, 4, 10, 23, 56, 132, ...) = first difference row of column 1. Left border of the triangle = A137682.

Left column starting (1, 3, ...) = INVERT transform of A160096. - Gary W. Adamson, May 01 2009

Examples

			First few rows of the triangle:
     1;
     1,   1;
     3,   0,   1;
     7,   2,   0,   1;
    17,   4,   1,   0,  1;
    40,  10,   4,   1,  0,  1;
    96,  23,   8,   3,  1,  0, 1;
   228,  56,  19,   8,  3,  1, 0, 1;
   544, 132,  46,  18,  7,  3, 1, 0, 1;
  1296, 316, 109,  42, 18,  7, 3, 1, 0, 1;
  3089, 752, 260, 101, 41, 17, 7, 3, 1, 0, 1;
  ...
		

Crossrefs

Cf. A137681 (row sums), A137682.

Cf. A160096. - Gary W. Adamson, May 01 2009

Programs

Maple

A137680 := proc(n,k)
    if k < 1 or k > n then
        0 ;
    elif n = 1 then
        1;
    elif k = 1 then
        add(add(procname(r,j),j=1..r),r=1..n-1) ;
    else
        procname(n-1,k-1)-procname(n-k,k-1) ;
    end if;
end proc: # R. J. Mathar, Aug 12 2012

Mathematica

T[n_, k_] := T[n, k] = Which[k < 1 || k > n, 0, n == 1, 1, k == 1, Sum[T[r, j], {r, 1, n-1}, {j, 1, r}], True, T[n-1, k-1] - T[n-k, k-1]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 02 2024, after R. J. Mathar *)