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Expansion of (1+2x-sqrt(1-4x^2))/(2(1-x^2)*sqrt(1-4x^2)).
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%I #16 Jul 18 2022 19:32:33

%S 0,1,1,3,4,9,14,29,49,99,175,351,637,1275,2353,4707,8788,17577,33098,

%T 66197,125476,250953,478192,956385,1830270,3660541,7030570,14061141,

%U 27088870,54177741,104647630

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

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

%C 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.

%H Vincenzo Librandi, <a href="/A174783/b174783.txt">Table of n, a(n) for n = 0..1000</a>

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

%F 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

%F a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 04 2014

%F a(n) = Sum_{i=1..(n+1)/2}((binomial(n-2*i+1,floor((n-2*i+1)/2)))). - _Vladimir Kruchinin_, Mar 15 2016

%t 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 *)

%o (Maxima)

%o a(n):=sum((binomial(n-2*i+1,floor((n-2*i+1)/2))),i,1,(n+1)/2); /* - _Vladimir Kruchinin_, Mar 15 2016 */

%Y Essentially partial sums of A086905.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Mar 29 2010