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%I #2 Jul 11 2010 03:00:00
%S 1,1,1,2,2,1,5,7,3,1,14,30,15,4,1,42,143,91,26,5,1,132,728,612,204,40,
%T 6,1,429,3876,4389,1771,385,57,7,1,1430,21318,32890,16380,4095,650,77,
%U 8,1,4862,120175,254475,158224,46376,8184,1015,100,9,1,16796,690690,2017356
%N Triangle, read by rows, defined by: T(n,k) = 1/((k+1)n-1) binomial((k+1)n-1,n) for n,k>0.
%C T(n,k) counts number of lattice paths with steps (1,k) and (1,-1) starting at the origin and ending at height 1 with i vertices on or below the x-axis for i=1,2,...,(r+1)n-1. For k=1, T(n,1) are the Catalan numbers A000108, k=2 gives the sequence A006013, k=3 gives the sequence A006632, k=4 gives the sequence A118971, etc.
%F Satisfies xf^k(x)=1-f^{-1}(x). Can also be written as T(n,k) = 1/n binomial((k+1)n-2,n-1) = 1/(kn-1) binomial((k+1)n-2,n)
%t TableForm[ Table[1/((k + 1) n - 1) Binomial[(k + 1) n - 1, n], {k, 1, 10}, {n, 1, 10}]]
%Y Cf. A000108, A006013, A006632, A118971, etc.
%K easy,nonn,tabl
%O 1,4
%A Aminul Huq (aminul(AT)brandeis.edu), Jul 02 2009
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