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A383474 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),(2,0),(3,0),(0,1),(0,2),(0,3).

%I A383474 #14 Apr 28 2025 08:45:21
%S A383474 1,1,1,2,2,2,4,5,5,4,7,12,14,12,7,13,26,37,37,26,13,24,56,89,106,89,
%T A383474 56,24,44,118,209,277,277,209,118,44,81,244,477,698,784,698,477,244,
%U A383474 81,149,499,1063,1700,2113,2113,1700,1063,499,149,274,1010,2329,4026,5469,6040,5469,4026,2329,1010,274
%N A383474 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),(2,0),(3,0),(0,1),(0,2),(0,3).
%F A383474 A(n,k) = A(k,n).
%F A383474 A(n,k) = A(n-1,k) + A(n-2,k) + A(n-3,k) + A(n,k-1) + A(n,k-2) + A(n,k-3).
%F A383474 G.f.: 1 / (1 - x - y - x^2 - y^2 - x^3 - y^3).
%e A383474 Square array A(n,k) begins:
%e A383474    1,   1,   2,    4,    7,    13,    24, ...
%e A383474    1,   2,   5,   12,   26,    56,   118, ...
%e A383474    2,   5,  14,   37,   89,   209,   477, ...
%e A383474    4,  12,  37,  106,  277,   698,  1700, ...
%e A383474    7,  26,  89,  277,  784,  2113,  5469, ...
%e A383474   13,  56, 209,  698, 2113,  6040, 16497, ...
%e A383474   24, 118, 477, 1700, 5469, 16497, 47332, ...
%o A383474 (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-x^3-y^3), n), k);
%Y A383474 Column k=0..1 give A000073(n+2), A073778(n+4).
%Y A383474 Main diagonal gives A122680.
%Y A383474 Cf. A007318, A036355, A383477.
%K A383474 nonn,tabl
%O A383474 0,4
%A A383474 _Seiichi Manyama_, Apr 27 2025