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A383552 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),(2,2).

%I A383552 #11 Apr 30 2025 09:11:40
%S A383552 1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,12,12,5,1,1,6,18,26,18,6,1,1,7,25,
%T A383552 47,47,25,7,1,1,8,33,76,101,76,33,8,1,1,9,42,114,189,189,114,42,9,1,1,
%U A383552 10,52,162,321,404,321,162,52,10,1,1,11,63,221,508,772,772,508,221,63,11,1
%N A383552 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),(2,2).
%F A383552 A(n,k) = A(k,n).
%F A383552 A(n,k) = A(n-1,k) + A(n,k-1) + A(n-2,k-2).
%F A383552 G.f.: 1 / (1 - x - y - x^2*y^2).
%e A383552 Square array A(n,k) begins:
%e A383552   1, 1,  1,   1,   1,   1,    1, ...
%e A383552   1, 2,  3,   4,   5,   6,    7, ...
%e A383552   1, 3,  7,  12,  18,  25,   33, ...
%e A383552   1, 4, 12,  26,  47,  76,  114, ...
%e A383552   1, 5, 18,  47, 101, 189,  321, ...
%e A383552   1, 6, 25,  76, 189, 404,  772, ...
%e A383552   1, 7, 33, 114, 321, 772, 1645, ...
%o A383552 (PARI) a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x-y-x^2*y^2), n), k);
%Y A383552 Main diagonal gives A349713.
%Y A383552 Cf. A008288, A383551, A383566.
%K A383552 nonn,tabl
%O A383552 0,5
%A A383552 _Seiichi Manyama_, Apr 30 2025