A039798 Triangle read by rows: numbers of Dyck paths.

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

Offset

0,5

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

Example

Triangle begins
  1;
  1,  1;
  1,  3,  3;
  1,  6, 14, 14;
  ...

Maple

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

Mathematica

Flatten[Table[(n+k)!(n+k+2)!(n-k+3)!/(k!(k+1)!(n-k)!(n+2)!(n+3)!), {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jul 16 2012 *)

Prog

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

Crossrefs

Cf. A039797.

Reflection of A039797.

Sequence in context: A219218 A208524 A094040 * A193560 A278390 A356916

Adjacent sequences: A039795 A039796 A039797 * A039799 A039800 A039801

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Apr 29 2004

Status

approved