A137896 Numerators of a rational triangle related to 1/sqrt(1-x).

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).

Links

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

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

Adjacent sequences: A137893 A137894 A137895 * A137897 A137898 A137899

Keyword

frac,nonn,tabl

Author

Paul Barry, Feb 21 2008

Status

approved