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A039797 Triangle of numbers of Dyck paths. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 8 08:31 EST 2021. Contains 349594 sequences. (Running on oeis4.)