A243022 Expansion of -(2*x)/(1-sqrt(1-(2*(1-sqrt(1-4*x^2)))/x)).

-1, 1, 2, 2, 7, 18, 61, 198, 694, 2446, 8873, 32556, 121243, 455986, 1731459, 6625258, 25527571, 98947914, 385587017, 1509702496, 5936181673, 23430706276, 92805006308, 368747893980, 1469408091637, 5870927247410

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

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.

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

a(n) ~ 4*(17/4)^n / (sqrt(255*Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 15 2014

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

Mathematica

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

Prog

(Maxima)
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);
(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

Crossrefs

Cf. A000108.

Sequence in context: A203176 A203579 A338415 * A049955 A184714 A156464

Adjacent sequences: A243019 A243020 A243021 * A243023 A243024 A243025

Keyword

sign

Author

Vladimir Kruchinin, Jun 09 2014

Status

approved