A249488 Square array A(n,k) for n,k>=0, where A(n,k) is the number of paths from (0,0) to (n,k) in the directed graph with vertices (i,j) and edges (i,j)-to-(i+1,j), (i,j)-to-(i,j+1), and (i,i+h)-to-(i+1,i+h+1) for every i,j,h>=0.

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 5, 12, 7, 1, 1, 6, 17, 24, 9, 1, 1, 7, 23, 53, 40, 11, 1, 1, 8, 30, 76, 117, 60, 13, 1, 1, 9, 38, 106, 246, 217, 84, 15, 1, 1, 10, 47, 144, 352, 580, 361, 112, 17, 1, 1, 11, 57, 191, 496, 1178, 1158, 557, 144, 19, 1, 1, 12, 68, 248, 687, 1674, 2916, 2076, 813, 180, 21, 1, 1, 13, 80, 316, 935, 2361, 5768, 6150, 3446, 1137, 220, 23, 1, 1, 14, 93

Offset: 0

Author

Max Alekseyev, Jan 13 2015

Keywords

• nonn
• tabl

Crossrefs

Row-reversed or transposed version of A026780.

Cf. A026781 (main diagonal), A026787 (sums of antidiagonals).

Formula

For n>=2*k, A(n,k) = coefficient of x^k in F(x)*C(x)^(n-2*k). For n<=2*k, A(n,k) = coefficient of x^(n-k) in F(x)*S(x)^(2*k-n). Here C(x)=(1-sqrt(1-4x))/(2*x) is o.g.f. for A000108, S(x)=(1-x-sqrt(1-6*x+x^2))/(2*x) is o.g.f. for A006318, and F(x)=S(x)/(1-x*C(x)*S(x)) is o.g.f. for A026781.