A243764 Expansion of -((2*sqrt(1-4*x)-2)*x)/(sqrt(8*x+4*sqrt(1-4*x)-3)-1).

-1, 1, 3, 4, 13, 36, 122, 408, 1453, 5236, 19374, 72616, 276194, 1061416, 4119060, 16112944, 63485565, 251700308, 1003515878, 4020985608, 16184539030, 65409684984, 265338949740, 1080038134352, 4409974794962, 18058663921992, 74146961705356, 305193422510224

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = (Sum_{k=0..floor(n/2)} 2^k*binomial(2*k-1,k)*binomial(2*n-2*k-2,n-2))/(n-1), n>1, a(0)=-1, a(1)=1.

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

a(n) ~ sqrt(19/sqrt(2)-3) * 2^(3*n-13/4) * ((1+2*sqrt(2))/7)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 15 2014

Conjecture: 245*n*(n-1)*(n-2)*a(n) -140*(n-1)*(n-2)*(28*n-81)*a(n-1) +4*(n-2)*(5812*n^2-41488*n+72261)*a(n-2) +16*(-3368*n^3+45144*n^2-191968*n+260859)*a(n-3) +128*(-152*n^3-768*n^2+16748*n-45483)*a(n-4) +2048*(2*n-11)*(74*n^2-715*n+1629)*a(n-5) -98304*(n-7)*(2*n-11)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jun 07 2016

Mathematica

CoefficientList[Series[-2*(-1 + Sqrt[1-4*x])*x/(-1 + Sqrt[-3 + 4*Sqrt[1-4*x] + 8*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(2^k*binomial(2*k-1, k)*binomial(2*n-2*k-2, n-2), k, 0, (n)/2)/(n-1);
(PARI) a(n) = if (n==0, -1, if (n==1, 1, sum(k=0, n\2, 2^k*binomial(2*k-1, k)*binomial(2*n-2*k-2, n-2))/(n-1))); \\ Michel Marcus, Jun 10 2014

Crossrefs

Cf. A000108.

Sequence in context: A084315 A194649 A062165 * A201821 A001056 A122151

Adjacent sequences: A243761 A243762 A243763 * A243765 A243766 A243767

Keyword

sign,changed

Author

Vladimir Kruchinin, Jun 10 2014

Status

approved