A141244 Numerators in the expansion of (1-sqrt(1-x^2))/(1-x).

0, 0, 1, 1, 5, 5, 11, 11, 93, 93, 193, 193, 793, 793, 1619, 1619, 26333, 26333, 53381, 53381, 215955, 215955, 436109, 436109, 3518265, 3518265, 7088533, 7088533, 28539857, 28539857, 57414019

Offset: 0

Author

Paul Barry, Jun 17 2008

Keywords

• easy
• frac
• nonn

Comments

The denominators in the expansion of (1-sqrt(1-x^2))/(1-x) are 1,1,2,2,8,8,16,16,... or 2^A005187(n) doubled. The sequence 0,1/2,1/2,5/8,5/8,... is the image of n under the Chebyshev related (rational) Riordan array c((x/2)^2),(x/2)c((x/2)^2)) with c(x) the g.f. of A000108. The image of n+1 under this array is 1,1,1,....

Links

• G. C. Chingas, A. N. Garroway, W. B. Moniz, and R. D. Bertrand, Adiabatic J cross-polarization in liquids for signal enhancement in NMR, Journal of Chemical Physics, 102:8 (1980), 2526-2528 (page 1, equation (2)).

Crossrefs

Cf. A000108, A005187, A120778.

Programs

PARI

a(n) = my(y=x+O(x^(n+1))); numerator(polcoeff((1-sqrt(1-y^2))/(1-y), n)); \\ Michel Marcus, Aug 17 2014

Formula

a(2n) = a(2n+1) = A120778(n-1) (conjectural). - Greg Martin, Aug 16 2014, corrected by M. F. Hasler, Aug 18 2014