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A387249 a(n) = 10/(n + 1) * Catalan(3*n).

%I A387249 #15 Sep 02 2025 11:01:15
%S A387249 10,25,440,12155,416024,16158075,682341000,30582833775,1433226830360,
%T A387249 69533550916004,3468169547356640,176946775343535925,
%U A387249 9199844912200348840,486018122664268428850,26029619941269629306160,1410698658798280045783575,77251704848334920869407000,4269325372507953547350453420
%N A387249 a(n) = 10/(n + 1) * Catalan(3*n).
%C A387249 Compare with Catalan(n) = 1/(n + 1) * binomial(2*n, n).
%C A387249 For r >= 2, there is a constant K_r such that K_r/(n + 1) * Catalan(r*n) is integral for all n.
%H A387249 Paolo Xausa, <a href="/A387249/b387249.txt">Table of n, a(n) for n = 0..500</a>
%F A387249 a(n) = 10/((n + 1)*(3*n + 1)) * binomial(6*n, 3*n).
%F A387249 a(n) = (3*n + 2)/2 * (16*Catalan(3*n) - 8*Catalan(3*n+1) + Catalan(3n+2)) (shows a(n) to be an integer since Catalan(n) is odd iff n = 2^k - 1 for some k, so Catalan(3*n+2) is always even).
%F A387249 a(n) = (3*n + 2)/2 * A007272(3*n).
%F A387249 a(n) = 8*(2*n - 1)*(6*n - 1)*(6*n - 5)/((n + 1)*(3*n + 1)*(3*n - 1)) * a(n-1) with a(0) = 10.
%F A387249 a(n) ~ 10/(sqrt(27*Pi)) * 64^n/n^(5/2).
%F A387249 E.g.f.: 10*hypergeom([1/6, 1/2, 5/6], [2/3, 4/3, 2], 64*x). - _Stefano Spezia_, Aug 27 2025
%p A387249 seq( 10/((n+1)*(3*n+1)) * binomial(6*n, 3*n), n = 0..20);
%t A387249 A387249[n_] := 10*CatalanNumber[3*n]/(n + 1); Array[A387249, 20, 0] (* _Paolo Xausa_, Sep 02 2025 *)
%Y A387249 Cf. A000108, A000984, A007272, A066802, A387248, A387250.
%K A387249 nonn,easy,new
%O A387249 0,1
%A A387249 _Peter Bala_, Aug 24 2025