A174783 Expansion of (1+2x-sqrt(1-4x^2))/(2(1-x^2)*sqrt(1-4x^2)).

0, 1, 1, 3, 4, 9, 14, 29, 49, 99, 175, 351, 637, 1275, 2353, 4707, 8788, 17577, 33098, 66197, 125476, 250953, 478192, 956385, 1830270, 3660541, 7030570, 14061141, 27088870, 54177741, 104647630

Offset: 0

Author

Paul Barry, Mar 29 2010

Keywords

• easy
• nonn

Comments

Hankel transform is A174784. Hankel transform of a(n+1) is A174785.

Transform of the sequence 0,1,1,1,1,0,0,1,1,1,1,0,0,1,.. by the Riordan array (c(x^2),xc(x^2)), c(x) the g.f. of A000108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Essentially partial sums of A086905.

Programs

Mathematica

CoefficientList[Series[(1 + 2 x - Sqrt[1 - 4 x^2])/(2 (1 - x^2) Sqrt[1 - 4 x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

Maxima

a(n):=sum((binomial(n-2*i+1,floor((n-2*i+1)/2))),i,1,(n+1)/2); /* - Vladimir Kruchinin, Mar 15 2016 */

Formula

E.g.f.: int(cosh(x-t)*(Bessel_I(0,2t)+Bessel_I(1,2t)),t,0,x).

Conjecture: n*a(n) -2*a(n-1) +(8-5*n)*a(n-2) +2*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Nov 13 2012

a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014

a(n) = Sum_{i=1..(n+1)/2}((binomial(n-2*i+1,floor((n-2*i+1)/2)))). - Vladimir Kruchinin, Mar 15 2016