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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A229730 A295193 A248097 * A220172 A276054 A192305

Adjacent sequences:  A098270 A098271 A098272 * A098274 A098275 A098276

KEYWORD

nonn,tabl,walk

AUTHOR

Ralf Stephan, Sep 02 2004

STATUS

approved

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Last modified August 18 04:50 EDT 2019. Contains 326072 sequences. (Running on oeis4.)