login
A137896
Numerators of a rational triangle related to 1/sqrt(1-x).
1
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 18, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 5, 100, 5, 6, 1, 1, 7, 63, 175, 175, 63, 7, 1, 1, 8, 28, 56, 490, 56, 28, 8, 1, 1, 9, 36, 84, 882, 882, 84, 36, 9, 1, 1, 10, 135, 120, 1470, 15876, 1470, 120, 135, 10, 1
OFFSET
0,5
COMMENTS
The rational triangle is the inverse of the coefficient array of the polynomial family defined by the sequence 1/(2n+1) (reflection coefficients). The polynomials are calculated by
p(n, x):=IF(n=0, 1, x*p(n-1,x)-a(n-1)*x^(n-1)*p(n-1,1/x)) where a(n)=1/(2n+1).
The row sums of the rational triangle are the reciprocals of the expansion of 1/sqrt(1-x).
EXAMPLE
Triangle begins
1,
1, 1,
1, 2, 1,
1, 3, 3, 1,
1, 4, 18, 4, 1,
1, 5, 10, 10, 5, 1,
1, 6, 5, 100, 5, 6, 1,
1, 7, 63, 175, 175, 63, 7, 1,
1, 8, 28, 56, 490, 56, 28, 8, 1,
1, 9, 36, 84, 882, 882, 84, 36, 9, 1,
1, 10, 135, 120, 1470, 15876, 1470, 120, 135, 10, 1
The associated rational triangle begins
1,
1,1,
1,2/3,1,
1,3/5,3/5,1,
1,4/7,18/35,4/7,1
CROSSREFS
Cf. A137897 (denominators).
Sequence in context: A229556 A159623 A143199 * A157219 A167040 A054450
KEYWORD
frac,nonn,tabl
AUTHOR
Paul Barry, Feb 21 2008
STATUS
approved