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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.
3

%I #15 Aug 16 2023 08:22:03

%S 1,1,4,1,8,28,1,12,64,212,1,16,116,520,1676,1,20,184,1052,4288,13604,

%T 1,24,268,1872,9316,35784,112380,1,28,368,3044,17976,81708,301440,

%U 940020,1,32,484,4632,31740,167376,713940,2558280,7936620,1,36,616,6700,52336,314932,1531000,6231100,21842560,67494980

%N 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.

%H R. J. Mathar, <a href="/A348595/a348595.pdf">Walks of up and right steps in the square lattice with blocked squares</a>

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

%e The array is symmetric; the non-redundant triangular part starts

%e 1

%e 1 4

%e 1 8 28

%e 1 12 64 212

%e 1 16 116 520 1676

%e 1 20 184 1052 4288 13604

%e 1 24 268 1872 9316 35784 112380

%e 1 28 368 3044 17976 81708 301440 940020

%e 1 32 484 4632 31740 167376 713940 2558280 7936620

%p A348595 := proc(n,k)

%p g := (1-u*v)/(1-u-v-3*u*v) ;

%p coeftayl(%,u=0,n) ;

%p coeftayl(%,v=0,k) ;

%p end proc:

%p seq(seq( A348595(n,k),k=0..n),n=0..10) ;

%t T[n_, k_] := Module[{u, v}, SeriesCoefficient[(1 - u v)/(1 - u - v - 3 u v), {u, 0, n}] // SeriesCoefficient[#, {v, 0, k}]&];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 16 2023 *)

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

%K nonn,tabl,easy

%O 0,3

%A _R. J. Mathar_, Jan 26 2022