OFFSET
0,5
COMMENTS
LINKS
Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 4.
FORMULA
Number triangle T(0, 0)=1, T(0, k)=0, k>0, T(n, k)=(k/n)*binomial(3n-k-1, n-k) otherwise; Riordan array (1, f) where f(1-f)^2=x.
T(n, k)=sum{j=0..n, ((3j+1)/(2n+j+1))(-1)^(j-k)*C(3n, 2n+j)C(j, k)}; - Paul Barry, Oct 07 2005
T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)
EXAMPLE
Rows begin
1;
0,1;
0,2,1;
0,7,4,1;
0,30,18,6,1;
0,143,88,33,8,1;
Production array begins
0, 1
0, 2, 1
0, 3, 2, 1
0, 4, 3, 2, 1
0, 5, 4, 3, 2, 1
0, 6, 5, 4, 3, 2, 1,
0, 7, 6, 5, 4, 3, 2, 1
0, 8, 7, 6, 5, 4, 3, 2, 1
0, 9, 8, 7, 6, 5, 4, 3, 2, 1
... - Philippe Deléham, Mar 05 2013
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 06 2005
STATUS
approved