A249512 - OEIS

%I #16 Oct 05 2022 08:38:28

%S 1,1,3,7,15,33,75,169,375,835,1875,4203,9375,20931,46875,104919,

%T 234375,523737,1171875,2621545,5859375,13098001,29296875,65523597,

%U 146484375,327500413,732421875,1637918089

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

%F a(n) = sum(k = 1..n, k*4^(n-k)*binomial(n/2,n-k))/n, a(0)=1.

%F a(n) ~ 3 * 5^(n/2-1). - _Vaclav Kotesovec_, Oct 31 2014

%F a(n) = 3 * 5^(n/2-1) if n is even and n>0 else a(n) = ((4^(n-1)* binomial(n/2, n-1)*hypergeometric([2, 1-n],[2-n/2], -1/4))/n). - _Peter Luschny_, Oct 31 2014

%F D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +(n+11)*a(n-2) +(n+10)*a(n-3) +20*(n-4)*a(n-4) +20*(n-5)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020

%p # Using function CompInv from A357588.

%p 1, CompInv(27, n -> simplify(GegenbauerC(n-1, 1-n, 3/2))); # _Peter Luschny_, Oct 05 2022

%t CoefficientList[Series[1/(1-x*Sqrt[4*x^2+1]-2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 31 2014 *)

%o (Maxima)

%o a(n) := if n=0 then 1  else sum(k*4^(n-k)*binomial(n/2,n-k),k,1,n)/n;

%o (Sage)

%o def a(n):

%o     if is_odd(n):

%o         return simplify((4^(n-1)*binomial(n/2, n-1)*hypergeometric([2, 1-n], [2-n/2], -1/4))/n)

%o     return 3*5^(n//2-1) if n>0 else 1

%o [a(n) for n in (0..27)] # _Peter Luschny_, Oct 31 2014

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Oct 31 2014