A348595 Triangle read by rows: Number of walks from (0,0) to (3n,3k) on the square lattice with up and right steps where squares (x,y)=(1,1) mod 3 or (x,y)=(2,2) mod 3 are not entered.

1, 1, 4, 1, 8, 28, 1, 12, 64, 212, 1, 16, 116, 520, 1676, 1, 20, 184, 1052, 4288, 13604, 1, 24, 268, 1872, 9316, 35784, 112380, 1, 28, 368, 3044, 17976, 81708, 301440, 940020, 1, 32, 484, 4632, 31740, 167376, 713940, 2558280, 7936620, 1, 36, 616, 6700, 52336, 314932, 1531000, 6231100, 21842560, 67494980

Offset: 0

Author

R. J. Mathar, Jan 26 2022

Keywords

• nonn
• tabl
• easy

Examples

			The array is symmetric; the non-redundant triangular part starts
   1
   1    4
   1    8   28
   1   12   64  212
   1   16  116  520  1676
   1   20  184 1052  4288  13604
   1   24  268 1872  9316  35784 112380
   1   28  368 3044 17976  81708 301440  940020
   1   32  484 4632 31740 167376 713940 2558280 7936620
		

Links

• R. J. Mathar, Walks of up and right steps in the square lattice with blocked squares

Crossrefs

Cf. A085363 (diagonal), A307584 (walks to (3n+1,3k))

Programs

Maple

A348595 := proc(n,k)
    g := (1-u*v)/(1-u-v-3*u*v) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,k) ;
end proc:
seq(seq( A348595(n,k),k=0..n),n=0..10) ;

Mathematica

T[n_, k_] := Module[{u, v}, SeriesCoefficient[(1 - u v)/(1 - u - v - 3 u v), {u, 0, n}] // SeriesCoefficient[#, {v, 0, k}]&];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

Formula

G.f.: (1-u*v)/(1-u-v-3*u*v) .