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A383551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,1),(3,3).

%I A383551 #9 Apr 30 2025 09:13:48
%S A383551 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,5,10,10,5,1,1,6,15,21,15,6,1,1,7,21,
%T A383551 37,37,21,7,1,1,8,28,59,76,59,28,8,1,1,9,36,88,138,138,88,36,9,1,1,10,
%U A383551 45,125,230,282,230,125,45,10,1,1,11,55,171,360,522,522,360,171,55,11,1
%N A383551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0),(0,1),(3,3).
%F A383551 A(n,k) = A(k,n).
%F A383551 A(n,k) = A(n-1,k) + A(n,k-1) + A(n-3,k-3).
%F A383551 G.f.: 1 / (1 - x - y - x^3*y^3).
%e A383551 Square array A(n,k) begins:
%e A383551   1, 1,  1,  1,   1,   1,    1, ...
%e A383551   1, 2,  3,  4,   5,   6,    7, ...
%e A383551   1, 3,  6, 10,  15,  21,   28, ...
%e A383551   1, 4, 10, 21,  37,  59,   88, ...
%e A383551   1, 5, 15, 37,  76, 138,  230, ...
%e A383551   1, 6, 21, 59, 138, 282,  522, ...
%e A383551   1, 7, 28, 88, 230, 522, 1065, ...
%o A383551 (PARI) a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x-y-x^3*y^3), n), k);
%Y A383551 Main diagonal gives A376791.
%K A383551 nonn,tabl
%O A383551 0,5
%A A383551 _Seiichi Manyama_, Apr 30 2025