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A098273 Array by antidiagonals: Number of planar lattice walks of length 3n+2k starting at (0,0) and ending at (k,0), remaining in the first quadrant and using only NE,W,S steps.

%I A098273 #21 Aug 30 2024 02:55:32
%S A098273 1,1,2,2,8,16,5,30,96,192,14,112,480,1408,2816,42,420,2240,8320,23296,
%T A098273 46592,132,1584,10080,44800,153600,417792,835584,429,6006,44352,
%U A098273 228480,913920,2976768,7938048,15876096,1430,22880,192192,1123584,5107200,19066880,59924480,157515776,315031552
%N A098273 Array by antidiagonals: Number of planar lattice walks of length 3n+2k starting at (0,0) and ending at (k,0), remaining in the first quadrant and using only NE,W,S steps.
%H A098273 G. Kreweras, <a href="http://www.numdam.org/numdam-bin/item?id=BURO_1965__6__9_0">Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), Eq. (85) p. 98.
%H A098273 M. Bousquet-Mélou, <a href="https://arxiv.org/abs/math/0401067">Walks in the quarter plane: Kreweras' algebraic model</a>, arXiv:math/0401067 [math.CO], 2004-2006.
%F A098273 T(n, k) = 4^n * (2k+1)/[(n+k+1)*(2n+2k+1)] * C(2k, k) * C(3n+2k, n).
%F A098273 T(n, k) = 2^(2*k)*(k+2*n)!/(k!*(2*n+2)!)*(2*n-2*k+2)!/((n-k)!*(n-k+1)!), as a triangle. - _Michel Marcus_, Nov 19 2014
%e A098273 As an array:
%e A098273   1    2    16    192    2816     46592
%e A098273   1    8    96   1408   23296    417792
%e A098273   2   30   480   8320  153600   2976768
%e A098273   5  112  2240  44800  913920  19066880
%e A098273   14 420 10080 228480 5107200 114250752
%e A098273   ...
%e A098273 As a regular triangle:
%e A098273   1;
%e A098273   1, 2;
%e A098273   2, 8, 16;
%e A098273   5, 30, 96, 192;
%e A098273   14, 112, 480, 1408, 2816;
%e A098273   ...
%t A098273 T[n_, k_] := 4^n (2k+1)/((n+k+1)(2n+2k+1)) Binomial[2k, k] Binomial[3n+2k, n];
%t A098273 Table[T[n-k, k], {n, 0, 8}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)
%o A098273 (PARI) T(n,k)=4^n*(2*k+1)/(n+k+1)/(2*n+2*k+1)*binomial(2*k,k)*binomial(3*n+2*k,n)
%o A098273 (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(2^(2*k)*(k+2*n)!/(k!*(2*n+2)!)*(2*n-2*k+2)!/((n-k)!*(n-k+1)!);, ", ");); print(););} \\ _Michel Marcus_, Nov 19 2014
%Y A098273 First row is A006335. First column is A000108 (Catalan numbers).
%K A098273 nonn,tabl,walk
%O A098273 0,3
%A A098273 _Ralf Stephan_, Sep 02 2004