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%I #20 Apr 02 2022 09:23:43
%S 2,10,140,2940,77616,2378376,80978040,2982691140,116776877360,
%T 4800591267472,205384736883872,9084324900632800,413286869105712000,
%U 19262120149391220000,916763612521908006000,44440565510927197408500,2189466044883038600910000
%N Product of 3 successive Catalan numbers.
%H Alois P. Heinz, <a href="/A014228/b014228.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ 64^(n+1) / (Pi^(3/2) * n^(9/2)). - _Vaclav Kotesovec_, Aug 25 2014
%F From _Amiram Eldar_, Apr 02 2022: (Start)
%F a(n) = C(n)*C(n+1)*C(n+2), where C(n) = A000108(n) is the n-th Catalan number.
%F Sum_{n>=0} a(n)/4^(3*n+3) = 1/2 - 64*Gamma(7/4)^2/(45*Pi*Gamma(5/4)^2). (End)
%p a:= proc(n) option remember; `if`(n=0, 2,
%p 8*(2*n-1)*(2*n+1)*(2*n+3) *a(n-1) /((n+1)*(n+2)*(n+3)))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Oct 20 2013
%t a[n_] := Times @@ CatalanNumber[{n, n+1, n+2}];
%t a /@ Range[0, 16] (* _Jean-François Alcover_, Dec 18 2020 *)
%Y Cf. A000108, A005568, A342288.
%K nonn
%O 0,1
%A _N. J. A. Sloane_.
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