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%I #24 Feb 21 2022 01:01:04
%S 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,
%T 2475,4719,4719,1,28,308,1925,7865,22022,40898,40898,1,36,504,4004,
%U 21021,78078,208208,379236,379236,1,45,780,7644,49686,231868,804440,2068560,3711916,3711916
%N Triangle read by rows: 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) = (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
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 3, 3;
%e 1, 6, 14, 14;
%e ...
%p 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);
%t 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 *)
%o (PARI) T(n, k)=(n+k)!*(n+k+2)!*(n-k+3)!/(k!*(k+1)!*(n-k)!*(n+2)!*(n+3)!);
%o matrix(10,10, n, k, if (n>=k, T(n-1,k-1))) \\ _Michel Marcus_, Mar 05 2020
%o (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
%Y Cf. A039797.
%Y Reflection of A039797.
%K nonn,tabl,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Emeric Deutsch_, Apr 29 2004
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