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A014228 Product of 3 successive Catalan numbers.

%I A014228 #22 Jul 08 2025 02:55:41
%S A014228 2,10,140,2940,77616,2378376,80978040,2982691140,116776877360,
%T A014228 4800591267472,205384736883872,9084324900632800,413286869105712000,
%U A014228 19262120149391220000,916763612521908006000,44440565510927197408500,2189466044883038600910000
%N A014228 Product of 3 successive Catalan numbers.
%H A014228 Alois P. Heinz, <a href="/A014228/b014228.txt">Table of n, a(n) for n = 0..300</a>
%F A014228 a(n) ~ 64^(n+1) / (Pi^(3/2) * n^(9/2)). - _Vaclav Kotesovec_, Aug 25 2014
%F A014228 From _Amiram Eldar_, Apr 02 2022: (Start)
%F A014228 a(n) = C(n)*C(n+1)*C(n+2), where C(n) = A000108(n) is the n-th Catalan number.
%F A014228 Sum_{n>=0} a(n)/4^(3*n+3) = 1/2 - 64*Gamma(7/4)^2/(45*Pi*Gamma(5/4)^2). (End)
%p A014228 a:= proc(n) option remember; `if`(n=0, 2,
%p A014228       8*(2*n-1)*(2*n+1)*(2*n+3) *a(n-1) /((n+1)*(n+2)*(n+3)))
%p A014228     end:
%p A014228 seq(a(n), n=0..20);  # _Alois P. Heinz_, Oct 20 2013
%t A014228 a[n_] := Times @@ CatalanNumber[{n, n+1, n+2}];
%t A014228 a /@ Range[0, 16] (* _Jean-François Alcover_, Dec 18 2020 *)
%Y A014228 Cf. A000108, A005568, A342288.
%K A014228 nonn
%O A014228 0,1
%A A014228 _N. J. A. Sloane_