A187914 Generalized Riordan array based on the binomial transform of the Fine's numbers A000957.

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, 96900, 171369, 74469, 52608, 17726, 8127, 2133, 612, 108, 15, 1

Offset

0,4

Comments

Row sums are A033321(n+1). Second column is A002212(n+1). Equal to A007318*A187913.

Links

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

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

Sequence in context: A289905 A086211 A110189 * A321625 A132372 A103136

Adjacent sequences: A187911 A187912 A187913 * A187915 A187916 A187917

Keyword

nonn,easy,tabl

Author

Paul Barry, Mar 15 2011

Status

approved