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
Links
- R. Kahkeshani, A Generalization of the Catalan Numbers, J. integer Seq., 16 (2013), Article 13.6.8.
Programs
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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
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Mathematica
Table[CatalanNumber[2n]Binomial[2n,n],{n, 0, 20}] (* Benedict W. J. Irwin, Oct 14 2016 *) -
PARI
a(n)=binomial(4*n,2*n)/(2*n+1) * binomial(2*n,n) \\ Charles R Greathouse IV, Oct 14 2016
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))!.
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
Comments