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7-dimensional Catalan numbers.
4

%I #24 Sep 08 2022 08:46:23

%S 1,1,429,1385670,13672405890,278607172289160,9490348077234178440,

%T 475073684264389879228560,32103104214166146088869942000,

%U 2760171874087743799855959353857200,289232890341906497299306268771988273600,35764585916110766978895474668714467232388000

%N 7-dimensional Catalan numbers.

%C Number of n X 7 Young tableaux.

%H Seiichi Manyama, <a href="/A321976/b321976.txt">Table of n, a(n) for n = 0..177</a>

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

%F a(n) = 0!*1!*...*6! * (7*n)! / ( n!*(n+1)!*...*(n+6)! ).

%F a(n) ~ 3110400 * 7^(7*n + 1/2) / (Pi^3 * n^24). - _Vaclav Kotesovec_, Nov 23 2018

%t Table[24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!),{n,0,15}] (* _Vincenzo Librandi_, Nov 24 2018 *)

%o (PARI) {a(n) = 24883200*(7*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!*(n+5)!*(n+6)!)}

%o (Magma) [24883200*Factorial(7*n) / (Factorial(n)*Factorial(n + 1)*Factorial(n + 2)*Factorial(n + 3)*Factorial(n + 4)*Factorial(n + 5)*Factorial(n + 6)): n in [0..15]]; // _Vincenzo Librandi_, Nov 24 2018

%o (GAP) List([0..15],n->24883200*Factorial(7*n)/Product([0..6],k->Factorial(n+k))); # _Muniru A Asiru_, Nov 25 2018

%Y Row 7 of A060854.

%Y Cf. A000108, A005789, A005790, A005791, A321975.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 23 2018