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}.

1, 4, 84, 2640, 100100, 4232592, 192203088, 9178678080, 455053212900, 23222793594000, 1212760632317520, 64534727833692480, 3488102039411078544, 191031492362224091200, 10580671081188491976000, 591771245038033007566080, 33380437374581432902637220

Offset

0,2

Comments

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

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.

Links

Table of n, a(n) for n=0..16.

R. Kahkeshani, A Generalization of the Catalan Numbers, J. integer Seq., 16 (2013), Article 13.6.8.

Formula

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

From Benedict W. J. Irwin, Oct 11 2016: (Start)

To clarify the above:

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

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

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

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

Maple

A270577 := proc(n)
    A000108(2*n)*binomial(2*n, n) ;
end proc:
seq(A270577(n), n=0..30) ; # R. J. Mathar, Jul 15 2017

Mathematica

Table[CatalanNumber[2n]Binomial[2n, n], {n, 0, 20}] (* Benedict W. J. Irwin, Oct 14 2016 *)

Prog

(PARI) a(n)=binomial(4*n, 2*n)/(2*n+1) * binomial(2*n, n) \\ Charles R Greathouse IV, Oct 14 2016

Crossrefs

Cf. A000108, A000984.

Sequence in context: A053352 A322330 A356504 * A156477 A024259 A357672

Adjacent sequences: A270574 A270575 A270576 * A270578 A270579 A270580

Keyword

nonn,easy

Author

Reza Kahkeshani, Mar 19 2016

Status

approved