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

0,3

Links

Table of n, a(n) for n=0..54.

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

Formula

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A366399 A100235 A089072 * A036177 A360131 A177841

Adjacent sequences: A348592 A348593 A348594 * A348596 A348597 A348598

Keyword

nonn,tabl,easy

Author

R. J. Mathar, Jan 26 2022

Status

approved