OFFSET
0,4
FORMULA
Let g(x)=(1+x-sqrt(1-6x+5x^2))/(2x(2-x)) be the g.f. of A033321, the binomial transform of the Fine numbers.
Then the g.f. of the k-th column is x^k*g(x)^((k+2)/2)/(1-2*x*g(x))^(k/2) if k is even, and
x^k*g(x)^((k+1)/2)/(1-2*x*g(x))^((k+1)/2) if k is odd. Otherwise put, column k has g.f.
g.f. x^k*g(x)^(k+1)/(1-xg(x)-x^2g(x)^2)^floor((k+1)/2).
EXAMPLE
Triangle begins
1,
1, 1,
2, 3, 1,
6, 10, 4, 1,
21, 36, 15, 6, 1,
79, 137, 58, 29, 7, 1,
311, 543, 232, 132, 37, 9, 1,
1265, 2219, 954, 590, 179, 57, 10, 1,
5275, 9285, 4010, 2628, 837, 315, 68, 12, 1,
22431, 39587, 17156, 11732, 3861, 1629, 396, 94, 13, 1
Production matrix is
1, 1,
1, 2, 1,
1, 2, 1, 1,
1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 1,
1, 2, 1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 2, 1, 1,
1, 2, 1, 2, 1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 2, 1, 2, 1, 1;
Hence, for instance, we have
79=1*0+1.21+1.36+1.15+1.6+1.1;
137=1.21+2.36+2.15+2.6+2.1;
58=1.36+1.15+1.6+1.1
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Mar 15 2011
STATUS
approved