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A270577 Generalized Catalan numbers C(3,n), where the (m,n)-th Catalan is the number of paths in R^m from the origin to the point (n,...,n,(m-1)n) with m kinds of moves such that the path never rises above the hyperplane x_m = x_1+...+x_{m-1}.

%I A270577 #40 Jul 31 2023 11:44:47
%S A270577 1,4,84,2640,100100,4232592,192203088,9178678080,455053212900,
%T A270577 23222793594000,1212760632317520,64534727833692480,
%U A270577 3488102039411078544,191031492362224091200,10580671081188491976000,591771245038033007566080,33380437374581432902637220
%N A270577 Generalized Catalan numbers C(3,n), where the (m,n)-th Catalan is the number of paths in R^m from the origin to the point (n,...,n,(m-1)n) with m kinds of moves such that the path never rises above the hyperplane x_m = x_1+...+x_{m-1}.
%C A270577 For any natural m>3, the other sequences can be obtained from C(m,n).
%C A270577 C(2,n) is the Catalan number C_n. Moreover, for example, C(4,1)=1, C(4,2)=11880, C(5,1)=336 and C(5,2)=3603600.
%H A270577 R. Kahkeshani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Kahkeshani/kahke3.html">A Generalization of the Catalan Numbers</a>, J. integer Seq., 16 (2013), Article 13.6.8.
%F A270577 C(m,n) = 1/(n(m-1)+1)*binomial(2n(m-1),n,...,n,n(m-1)).
%F A270577 From _Benedict W. J. Irwin_, Oct 11 2016: (Start)
%F A270577 To clarify the above:
%F A270577 C(m,n) = 1/(n*(m-1)+1)*(2*n*(m-1))!/(n!)^(m-1)/(n*(m-1))!.
%F A270577 a(n) = C(3,n) = Catalan(2*n) * binomial(2*n,n) = A000108(2*n)*A000984(n).
%F A270577 G.f.: 3F2(1/4,1/2,3/4; 1,3/2; 64*x). (End)
%F A270577 n^2*(2*n+1)*a(n) +4*-(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - _R. J. Mathar_, Jul 15 2017
%p A270577 A270577 := proc(n)
%p A270577     A000108(2*n)*binomial(2*n,n) ;
%p A270577 end proc:
%p A270577 seq(A270577(n),n=0..30) ; # _R. J. Mathar_, Jul 15 2017
%t A270577 Table[CatalanNumber[2n]Binomial[2n,n],{n, 0, 20}] (* _Benedict W. J. Irwin_, Oct 14 2016 *)
%o A270577 (PARI) a(n)=binomial(4*n,2*n)/(2*n+1) * binomial(2*n,n) \\ _Charles R Greathouse IV_, Oct 14 2016
%Y A270577 Cf. A000108, A000984.
%K A270577 nonn,easy
%O A270577 0,2
%A A270577 _Reza Kahkeshani_, Mar 19 2016