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Multi-dimensional Catalan numbers: diagonal T(n,n+2) of A060854.
3

%I #22 Jun 23 2022 12:03:08

%S 1,14,6006,140229804,278607172289160,67867669180627125604080,

%T 2760171874087743799855959353857200,

%U 24486819823897171791550434989846505231774984000,59986874261544072491135645330451363110127974096720977464312000

%N Multi-dimensional Catalan numbers: diagonal T(n,n+2) of A060854.

%H G. C. Greubel, <a href="/A060856/b060856.txt">Table of n, a(n) for n = 1..29</a>

%F a(n) = 0!*1!*...*(k-1)! *(k*n)! / ( n!*(n+1)!*...*(n+k-1)! ) for k=n+2.

%F a(n) ~ sqrt(Pi) * exp(n^2/2 + 2*n + 25/12) * n^(n^2 + 2*n + 11/12) / (A * 2^(2*n^2 + 4*n + 17/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 09 2015

%F a(n) = A039622(n+1) / (n+1). - _Tom Copeland_, May 30 2022

%t Table[Product[j!/(n+j)!,{j,0,n+1}]*(n*(n+2))!,{n,1,10}] (* _Vaclav Kotesovec_, Mar 09 2015 *)

%Y Cf. A060854, A074962.

%Y Cf. A039622.

%K nonn,easy

%O 1,2

%A _R. H. Hardin_, May 03 2001