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

0,7

Comments

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

Links

Table of n, a(n) for n=0..65.

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).

Example

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

Crossrefs

Sequence in context: A201758 A096110 A207260 * A329458 A332010 A201287

Adjacent sequences: A187910 A187911 A187912 * A187914 A187915 A187916

Keyword

nonn,easy,tabl

Author

Paul Barry, Mar 15 2011

Status

approved