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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 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.)