A026670 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for n >= 0; for n >= 1, T(n,1) = T(n,n-1) = n+1; for n >= 2, T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-1,k) if n is even and k = n/2, else T(n,k) = T(n-1,k-1) + T(n-1,k).

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 11, 5, 1, 1, 6, 16, 16, 6, 1, 1, 7, 22, 43, 22, 7, 1, 1, 8, 29, 65, 65, 29, 8, 1, 1, 9, 37, 94, 173, 94, 37, 9, 1, 1, 10, 46, 131, 267, 267, 131, 46, 10, 1, 1, 11, 56, 177, 398, 707, 398, 177, 56, 11, 1, 1, 12, 67

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			E.g., 11 = T(4, 2) = T(3, 1) + T(2, 2) + T(3, 2) = 4 + 3 + 4.
Triangle begins:
1
1  1
1  3  1
1  4  4   1
1  5 11   5   1
1  6 16  16   6    1
1  7 22  43  22    7    1
1  8 29  65  65   29    8   1
1  9 37  94 173   94   37   9   1
1 10 46 131 267  267  131  46  10  1
1 11 56 177 398  707  398 177  56 11  1
1 12 67 233 575 1105 1105 575 233 67 12 1
... - _Philippe Deléham_, Feb 02 2014
		

Links

• Rob Arthan, Comments on A026674, A026725, A026670

Crossrefs

Cf. A026674.

Formula

T(n, k) = number of paths from (0, 0) to (n-k, k) in the directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i=j.

Extensions

Formula corrected by David Perkinson (davidp(AT)reed.edu), Sep 19 2001 and also by Rob Arthan, Jan 16 2003

Typo in name corrected by Sean A. Irvine, Oct 09 2019

Offset corrected by R. J. Mathar and Sean A. Irvine, Oct 25 2019