A039797 Triangle of numbers of Dyck paths.

1, 1, 1, 3, 3, 1, 14, 14, 6, 1, 84, 84, 40, 10, 1, 594, 594, 300, 90, 15, 1, 4719, 4719, 2475, 825, 175, 21, 1, 40898, 40898, 22022, 7865, 1925, 308, 28, 1, 379236, 379236, 208208, 78078, 21021, 4004, 504, 36, 1, 3711916, 3711916, 2068560, 804440, 231868, 49686, 7644, 780, 45, 1

Offset

0,4

Links

Table of n, a(n) for n=0..54.

D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableau de Young, pp. 112-125 of "Combinatoire Enumérative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

Index entries for sequences related to Young tableaux.

Formula

T(n, k) = (2n-k)!*(2n-k+2)!*(k+3)!/((n-k)!*(n-k+1)!*k!*(n+2)!*(n+3)!) for 0 <= k <= n. - Emeric Deutsch, Apr 29 2004

Example

Triangle begins:
     1,
     1,  1,
     3,  3,     1,
    14,  14,    6,   1,
    84,  84,   40,  10,   1,
   594, 594,  300,  90,  15,  1,
  4719, 4719, 2475, 825, 175, 21, 1,
...

Maple

T:=(n, k)->(2*n-k)!*(2*n-k+2)!*(k+3)!/(n-k)!/(n-k+1)!/k!/(n+2)!/(n+3)!: seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

Flatten[Table[((2n-k)!(2n-k+2)!(k+3)!)/((n-k)!(n-k+1)!k!(n+2)!(n+3)!), {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jan 27 2012 *)

Prog

(PARI) T(n, k) = (2*n-k)!*(2*n-k+2)!*(k+3)!/((n-k)!*(n-k+1)!*k!*(n+2)!*(n+3)!);
matrix(8, 8, n, k, if (n>=k, T(n-1, k-1))) \\ Michel Marcus, Mar 05 2020
(Magma) /* As triangle */ [[Factorial(2*n - k) * Factorial(2*n - k + 2) * Factorial(k + 3) / (Factorial(n - k) * Factorial(n - k + 1) * Factorial(k) * Factorial(n + 2) * Factorial(n + 3)): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Mar 06 2020

Crossrefs

Reflection of A039798.

Sequence in context: A115193 A227343 A216294 * A143171 A112292 A001497

Adjacent sequences: A039794 A039795 A039796 * A039798 A039799 A039800

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Apr 29 2004

Status

approved