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A065424 Catalan-like formula: a(n) = binomial(6*n, 3*n+1)/(9*n+6).

%I A065424 #30 Sep 04 2025 01:53:38
%S A065424 1,33,1326,59432,2851425,143291610,7446255180,396893583792,
%T A065424 21579377870484,1192183281903845,66734212415276406,
%U A065424 3776778437640143208,215744630060724034270,12423227699242323077940,720356761939547257421400,42024927437494196952957408,2464931252806478840545733484
%N A065424 Catalan-like formula: a(n) = binomial(6*n, 3*n+1)/(9*n+6).
%H A065424 Vincenzo Librandi, <a href="/A065424/b065424.txt">Table of n, a(n) for n = 1..100</a>
%F A065424 G.f.: A*sqrt((A+1)*(1+9*A)) where A=x*(1+9*A)^3*(1+A). - _Mark van Hoeij_, Nov 16 2011
%F A065424 -(n-1)*(3*n+2)*(3*n+1)*a(n) + 8*(6*n-5)*(6*n-1)*(2*n-1)*a(n-1) = 0. - _R. J. Mathar_, Oct 31 2015
%F A065424 a(n) = (1/9)*A000245(3*n) = n*A000108(3*n)/(3*n + 2) for n >= 1. - _Peter Bala_, Mar 08 2023
%F A065424 a(n) ~ 2^(6*n) / (9 * sqrt(3*Pi) * n^(3/2)). - _Amiram Eldar_, Sep 04 2025
%t A065424 a[n_] := Binomial[6*n, 3*n+1]/(9*n+6); Array[a, 20] (* _Amiram Eldar_, Sep 04 2025 *)
%o A065424 (Magma) [Binomial(6*n,3*n+1)/(9*n+6): n in [1..20]]; // _Vincenzo Librandi_, Nov 17 2011
%Y A065424 Cf. A000108, A000245, A065097.
%K A065424 nonn,easy,changed
%O A065424 1,2
%A A065424 _Len Smiley_, Nov 16 2001