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.

1, 1, 2, 2, 8, 16, 5, 30, 96, 192, 14, 112, 480, 1408, 2816, 42, 420, 2240, 8320, 23296, 46592, 132, 1584, 10080, 44800, 153600, 417792, 835584, 429, 6006, 44352, 228480, 913920, 2976768, 7938048, 15876096, 1430, 22880, 192192, 1123584, 5107200, 19066880, 59924480, 157515776, 315031552

Offset

0,3

Links

Table of n, a(n) for n=0..44.

G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), Eq. (85) p. 98.

M. Bousquet-Mélou, Walks in the quarter plane: Kreweras' algebraic model, arXiv:math/0401067 [math.CO], 2004-2006.

Formula

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

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

Example

As an array:
  1    2    16    192    2816     46592
  1    8    96   1408   23296    417792
  2   30   480   8320  153600   2976768
  5  112  2240  44800  913920  19066880
  14 420 10080 228480 5107200 114250752
  ...
As a regular triangle:
  1;
  1, 2;
  2, 8, 16;
  5, 30, 96, 192;
  14, 112, 480, 1408, 2816;
  ...

Mathematica

T[n_, k_] := 4^n (2k+1)/((n+k+1)(2n+2k+1)) Binomial[2k, k] Binomial[3n+2k, n];
Table[T[n-k, k], {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 25 2018 *)

Prog

(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)
(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

Crossrefs

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

Sequence in context: A349648 A295193 A248097 * A342835 A361294 A220172

Adjacent sequences: A098270 A098271 A098272 * A098274 A098275 A098276

Keyword

nonn,tabl,walk

Author

Ralf Stephan, Sep 02 2004

Status

approved