A137897 Denominators of a rational triangle related to 1/sqrt(1-x).

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 35, 7, 1, 1, 9, 21, 21, 9, 1, 1, 11, 11, 231, 11, 11, 1, 1, 13, 143, 429, 429, 143, 13, 1, 1, 15, 65, 143, 1287, 143, 65, 15, 1, 1, 17, 85, 221, 2431, 2431, 221, 85, 17, 1, 1, 19, 323, 323, 4199, 46189, 4199, 323

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..62.

Example

Triangle begins
1,
1, 1,
1, 3, 1,
1, 5, 5, 1,
1, 7, 35, 7, 1,
1, 9, 21, 21, 9, 1,
1, 11, 11, 231, 11, 11, 1,
1, 13, 143, 429, 429, 143, 13, 1,
1, 15, 65, 143, 1287, 143, 65, 15, 1,
1, 17, 85, 221, 2431, 2431, 221, 85, 17, 1,
1, 19, 323, 323, 4199, 46189, 4199, 323, 323, 19, 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. A137896 (numerators).

Sequence in context: A338934 A228356 A253670 * A296327 A347970 A296541

Adjacent sequences: A137894 A137895 A137896 * A137898 A137899 A137900

Keyword

frac,nonn,tabl

Author

Paul Barry, Feb 21 2008

Status

approved