A039797 - OEIS

%I #25 Feb 21 2022 01:01:17

%S 1,1,1,3,3,1,14,14,6,1,84,84,40,10,1,594,594,300,90,15,1,4719,4719,

%T 2475,825,175,21,1,40898,40898,22022,7865,1925,308,28,1,379236,379236,

%U 208208,78078,21021,4004,504,36,1,3711916,3711916,2068560,804440,231868,49686,7644,780,45,1

%N Triangle of numbers of Dyck paths.

%H D. Gouyou-Beauchamps, <a href="https://doi.org/10.1007/BFb0072513">Chemins sous-diagonaux et tableau de Young</a>, pp. 112-125 of "Combinatoire Enumérative (Montreal 1985)", Lect. Notes Math. 1234, 1986.

%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>

%F 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

%e Triangle begins:

%e      1,

%e      1,  1,

%e      3,  3,     1,

%e     14,  14,    6,   1,

%e     84,  84,   40,  10,   1,

%e    594, 594,  300,  90,  15,  1,

%e   4719, 4719, 2475, 825, 175, 21, 1,

%e ...

%p 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);

%t 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 *)

%o (PARI) T(n, k) = (2*n-k)!*(2*n-k+2)!*(k+3)!/((n-k)!*(n-k+1)!*k!*(n+2)!*(n+3)!);

%o matrix(8, 8, n, k, if (n>=k, T(n-1,k-1))) \\ _Michel Marcus_, Mar 05 2020

%o (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

%Y Reflection of A039798.

%K nonn,tabl,easy,nice

%O 0,4

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, Apr 29 2004