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A119365 Generalized Catalan numbers for triangle A119335.

%I A119365 #15 Sep 14 2023 13:44:58
%S A119365 1,0,0,1,6,20,51,126,392,1513,5877,21054,71270,242463,863590,3193737,
%T A119365 11889414,43783908,159998493,586908936,2175907284,8138471667,
%U A119365 30541703733,114620380032,430344635913,1619584557885,6116422089050
%N A119365 Generalized Catalan numbers for triangle A119335.
%C A119365 Counts rooted planar n-trees whose number of leaves is divisible by 3.
%F A119365 a(n) = A119335(2n,n) - A119335(2n,n+1).
%F A119365 a(n) = Sum_{k=0..n} if(mod(n-k,3)=0, (1/n)*C(n,k)*C(n,k+1), 0).
%F A119365 a(n) + A119366(n) + A119367(n) = A000108(n).
%p A119365 A119365 := proc(n)
%p A119365     local k;
%p A119365     if n = 0 then
%p A119365         return 1
%p A119365     end if;
%p A119365     a := 0 ;
%p A119365     for k from 0 to n do
%p A119365         if modp(n-k,3) = 0 then
%p A119365             a := a+binomial(n,k)*binomial(n,k+1) ;
%p A119365         end if;
%p A119365     end do:
%p A119365     a/n;
%p A119365 end proc:
%p A119365 seq(A119365(n),n=0..40) ; # _R. J. Mathar_, Oct 30 2014
%t A119365 A119335[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
%t A119365 a[n_] := A119335[2n, n] - A119335[2n, n+1];
%t A119365 Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Sep 14 2023 *)
%Y A119365 Cf. A000108, A001263, A119335, A119366, A119367.
%K A119365 easy,nonn
%O A119365 0,5
%A A119365 _Paul Barry_, May 16 2006