A321975 - OEIS

%I #27 Sep 08 2022 08:46:23

%S 1,1,132,87516,140229804,396499770810,1671643033734960,

%T 9490348077234178440,67867669180627125604080,

%U 583692803893929928888544400,5838544419011620940996212276800,66244124978105851196543024492572800,836288764382254532915188713779640302400,11570895443447601081407359451642915869302000

%N 6-dimensional Catalan numbers.

%C Number of n X 6 Young tableaux.

%H Seiichi Manyama, <a href="/A321975/b321975.txt">Table of n, a(n) for n = 0..222</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hook_length_formula">Hook length formula</a>

%F a(n) = 0!*1!*...*5! * (6*n)! / ( n!*(n+1)!*...*(n+5)! ).

%F a(n) ~ 5 * 2^(6*n + 6) * 3^(6*n + 7/2) / (Pi^(5/2) * n^(35/2)). - _Vaclav Kotesovec_, Nov 23 2018

%p a:= n-> (6*n)! * mul(i!/(6+i)!, i=0..n-1):

%p seq(a(n), n=0..14);  # _Alois P. Heinz_, Nov 25 2018

%t Table[34560 (6 n)! / (n! (n + 1)! (n + 2)! (n + 3)! (n + 4)! (n + 5)!), {n, 0, 60}] (* _Vincenzo Librandi_, Nov 24 2018 *)

%o (PARI) {a(n) = 34560*(6*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!)}

%o (Magma) [34560*Factorial(6*n)/(Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)): n in [0..15]]; // _Vincenzo Librandi_, Nov 24 2018

%o (GAP) List([0..15],n->34560*Factorial(6*n)/Product([0..5],k->Factorial(n+k))); # _Muniru A Asiru_, Nov 25 2018

%Y Row 6 of A060854.

%Y Cf. A000108, A005789, A005790, A005791.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 23 2018