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A383477 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).

%I A383477 #16 Apr 28 2025 08:45:11
%S A383477 1,1,1,1,2,2,1,3,5,4,1,4,9,12,7,1,5,14,25,26,13,1,6,20,44,63,56,24,1,
%T A383477 7,27,70,125,153,118,44,1,8,35,104,220,336,359,244,81,1,9,44,147,357,
%U A383477 646,864,819,499,149,1,10,54,200,546,1134,1800,2144,1830,1010,274
%N A383477 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).
%F A383477 A(n,k) = A(n-1,k) + A(n-2,k) + A(n-3,k) + A(n,k-1).
%F A383477 G.f.: 1 / (1 - x - x^2 - x^3 - y).
%F A383477 G.f. of column k: 1 / (1 - x - x^2 - x^3)^(k+1).
%e A383477 Square array A(n,k) begins:
%e A383477    1,   1,   1,   1,    1,    1,    1, ...
%e A383477    1,   2,   3,   4,    5,    6,    7, ...
%e A383477    2,   5,   9,  14,   20,   27,   35, ...
%e A383477    4,  12,  25,  44,   70,  104,  147, ...
%e A383477    7,  26,  63, 125,  220,  357,  546, ...
%e A383477   13,  56, 153, 336,  646, 1134, 1862, ...
%e A383477   24, 118, 359, 864, 1800, 3395, 5950, ...
%o A383477 (PARI) a(n, k) = my(x='x+O('x^(n+1)), y='y+O('y^(k+1))); polcoef(polcoef(1/(1-x-x^2-x^3-y), n), k);
%Y A383477 Column k=0..2 give A000073(n+2), A073778(n+4), A292326(n-1).
%Y A383477 Main diagonal gives A383478.
%Y A383477 Cf. A038137, A383474.
%K A383477 nonn,tabl
%O A383477 0,5
%A A383477 _Seiichi Manyama_, Apr 28 2025