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A039798 Triangle read by rows: numbers of Dyck paths. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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 A001498

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

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)