|
|
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
|
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|