A270577 - OEIS

%I #40 Jul 31 2023 11:44:47

%S 1,4,84,2640,100100,4232592,192203088,9178678080,455053212900,

%T 23222793594000,1212760632317520,64534727833692480,

%U 3488102039411078544,191031492362224091200,10580671081188491976000,591771245038033007566080,33380437374581432902637220

%N 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 For any natural m>3, the other sequences can be obtained from C(m,n).

%C 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 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 C(m,n) = 1/(n(m-1)+1)*binomial(2n(m-1),n,...,n,n(m-1)).

%F From _Benedict W. J. Irwin_, Oct 11 2016: (Start)

%F To clarify the above:

%F C(m,n) = 1/(n*(m-1)+1)*(2*n*(m-1))!/(n!)^(m-1)/(n*(m-1))!.

%F a(n) = C(3,n) = Catalan(2*n) * binomial(2*n,n) = A000108(2*n)*A000984(n).

%F G.f.: 3F2(1/4,1/2,3/4; 1,3/2; 64*x). (End)

%F 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 := proc(n)

%p     A000108(2*n)*binomial(2*n,n) ;

%p end proc:

%p seq(A270577(n),n=0..30) ; # _R. J. Mathar_, Jul 15 2017

%t Table[CatalanNumber[2n]Binomial[2n,n],{n, 0, 20}] (* _Benedict W. J. Irwin_, Oct 14 2016 *)

%o (PARI) a(n)=binomial(4*n,2*n)/(2*n+1) * binomial(2*n,n) \\ _Charles R Greathouse IV_, Oct 14 2016

%Y Cf. A000108, A000984.

%K nonn,easy

%O 0,2

%A _Reza Kahkeshani_, Mar 19 2016