A187913 Generalized Riordan array based on the Fine's numbers A000957.

1, 0, 1, 1, 1, 1, 2, 4, 1, 1, 6, 10, 5, 2, 1, 18, 32, 13, 9, 2, 1, 57, 100, 44, 28, 10, 3, 1, 186, 329, 142, 100, 32, 15, 3, 1, 622, 1101, 480, 344, 119, 55, 16, 4, 1, 2120, 3761, 1640, 1214, 420, 216, 60, 22, 4, 1, 7338, 13035, 5698, 4300, 1517, 810, 243, 92, 23, 5, 1

Offset: 0

Author

Paul Barry, Mar 15 2011

Keywords

• nonn
• easy
• tabl

Comments

First column is the Fine's numbers A000957. Row sums are A000958. Inverse binomial transform of A187914.

Examples

			Triangle begins
1,
0, 1,
1, 1, 1,
2, 4, 1, 1,
6, 10, 5, 2, 1,
18, 32, 13, 9, 2, 1,
57, 100, 44, 28, 10, 3, 1,
186, 329, 142, 100, 32, 15, 3, 1,
622, 1101, 480, 344, 119, 55, 16, 4, 1,
2120, 3761, 1640, 1214, 420, 216, 60, 22, 4, 1,
7338, 13035, 5698, 4300, 1517, 810, 243, 92, 23, 5, 1
Production matrix is
0, 1,
1, 1, 1,
1, 2, 0, 1,
1, 2, 1, 1, 1,
1, 2, 1, 2, 0, 1,
1, 2, 1, 2, 1, 1, 1,
1, 2, 1, 2, 1, 2, 0, 1,
1, 2, 1, 2, 1, 2, 1, 1, 1,
1, 2, 1, 2, 1, 2, 1, 2, 0, 1
Thus
57=1.0+0.18+1.32+1.13+1.9+1.2+1.1;
100=1.18+1.32+2.13+2.9+2.2+2.1;
44=1.32+0.13+1.9+1.2+1.1
		

Formula

Let g(x)=(1+2x-sqrt(1-4x))/(2x(2+x)) be the g.f. of the Fine's numbers A000957. Then column k has

g.f. x^k*g(x)^(k+1)/(1-xg(x)-x^2g(x)^2)^floor((k+1)/2).