A098273 - OEIS

%I #21 Aug 30 2024 02:55:32

%S 1,1,2,2,8,16,5,30,96,192,14,112,480,1408,2816,42,420,2240,8320,23296,

%T 46592,132,1584,10080,44800,153600,417792,835584,429,6006,44352,

%U 228480,913920,2976768,7938048,15876096,1430,22880,192192,1123584,5107200,19066880,59924480,157515776,315031552

%N 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 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 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 T(n, k) = 4^n * (2k+1)/[(n+k+1)*(2n+2k+1)] * C(2k, k) * C(3n+2k, n).

%F 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 As an array:

%e   1    2    16    192    2816     46592

%e   1    8    96   1408   23296    417792

%e   2   30   480   8320  153600   2976768

%e   5  112  2240  44800  913920  19066880

%e   14 420 10080 228480 5107200 114250752

%e   ...

%e As a regular triangle:

%e   1;

%e   1, 2;

%e   2, 8, 16;

%e   5, 30, 96, 192;

%e   14, 112, 480, 1408, 2816;

%e   ...

%t T[n_, k_] := 4^n (2k+1)/((n+k+1)(2n+2k+1)) Binomial[2k, k] Binomial[3n+2k, n];

%t Table[T[n-k, k], {n, 0, 8}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jul 25 2018 *)

%o (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 (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 First row is A006335. First column is A000108 (Catalan numbers).

%K nonn,tabl,walk

%O 0,3

%A _Ralf Stephan_, Sep 02 2004