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A187914
Generalized Riordan array based on the binomial transform of the Fine's numbers A000957.
1
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.
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
KEYWORD
nonn,easy,tabl
AUTHOR
Paul Barry, Mar 15 2011
STATUS
approved