A243022 - OEIS

%I #29 Jan 30 2020 21:29:17

%S -1,1,2,2,7,18,61,198,694,2446,8873,32556,121243,455986,1731459,

%T 6625258,25527571,98947914,385587017,1509702496,5936181673,

%U 23430706276,92805006308,368747893980,1469408091637,5870927247410

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

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

%F G.f.: A(x) =-1/(C(x^2)*C(x*C(x^2))), where C(x) is g.f. of A000108.

%F a(n) ~ 4*(17/4)^n / (sqrt(255*Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jun 15 2014

%F Conjecture D-finite with recurrence: +2*n*(n-1)*(2*n-3)*(5*n-12)*a(n) -(n-1)*(85*n^3-459*n^2+776*n-400)*a(n-1) +4*(-40*n^4+396*n^3-1455*n^2+2324*n-1360)*a(n-2) +4*(n-4)*(170*n^3-1258*n^2+3003*n-2185)*a(n-3) +16*(n-4)*(n-5)*(20*n^2-118*n+143)*a(n-4) -272*(n-4)*(n-5)*(n-6)*(5*n-7)*a(n-5)=0. - _R. J. Mathar_, Jul 15 2017

%t CoefficientList[Series[2*x/(-1 + Sqrt[(-2 + x + 2*Sqrt[1-4*x^2])/x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jun 15 2014 *)

%o (Maxima)

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

%o (PARI) a(n) = if (n==0, -1, if (n==1, 1, sum(k=0, n\2, binomial(n-1, k)*binomial(2*n-4*k-2, n-2*k))/(n-1))); \\ _Michel Marcus_, Jun 10 2014

%Y Cf. A000108.

%K sign

%O 0,3

%A _Vladimir Kruchinin_, Jun 09 2014