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

0,5

Links

Table of n, a(n) for n=0..68.

Rob Arthan, Comments on A026674, A026725, A026670

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.

Example

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

Crossrefs

Cf. A026674.

Sequence in context: A050177 A013580 A147290 * A131402 A238498 A026626

Adjacent sequences: A026667 A026668 A026669 * A026671 A026672 A026673

Keyword

nonn,tabl

Author

Clark Kimberling

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

Status

approved