A162382 Triangle, read by rows, defined by: T(n,k) = 1/((k+1)n-1) binomial((k+1)n-1,n) for n,k>0.

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, 6, 1, 429, 3876, 4389, 1771, 385, 57, 7, 1, 1430, 21318, 32890, 16380, 4095, 650, 77, 8, 1, 4862, 120175, 254475, 158224, 46376, 8184, 1015, 100, 9, 1, 16796, 690690, 2017356

Offset: 1

Author

Aminul Huq (aminul(AT)brandeis.edu), Jul 02 2009

Keywords

• easy
• nonn
• tabl

Comments

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.

Crossrefs

Cf. A000108, A006013, A006632, A118971, etc.

Programs

Mathematica

TableForm[ Table[1/((k + 1) n - 1) Binomial[(k + 1) n - 1, n], {k, 1, 10}, {n, 1, 10}]]

Formula

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)