A243760 Expansion of (sqrt(8*x+4*sqrt(1-4*x)-3)-1)/(2*sqrt(1-4*x)-2).

0, 1, 1, 4, 11, 40, 138, 512, 1903, 7256, 27910, 108704, 426702, 1688080, 6719252, 26895360, 108171319, 436935544, 1771673454, 7208637920, 29422782282, 120436341360, 494277785356, 2033457590656, 8384379887334, 34642507651952, 143413034719036, 594776592764224

Offset

0,4

Links

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

Formula

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

G.f. A(x) = x*C(x)*C(2*x^2*C(x)^2), where C(x) is the g.f. of A000108.

G.f. A(x) satisfies A(x)= x*(4*A(x)^4+4*A(x)^2+1)/(2*A(x)^2-A(x)+1).

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

Conjecture D-finite with recurrence: 245*n*(n-1)*(n+1)*a(n) -140*n*(n-1)*(28*n-53)*a(n-1) +4*(n-1)*(5812*n^2-29864*n+36585)*a(n-2) +16*(-3368*n^3+35040*n^2-111784*n+110667)*a(n-3) +128*(-152*n^3-1224*n^2+14756*n-29655)*a(n-4) +2048*(2*n-9)*(74*n^2-567*n+988)*a(n-5) -98304*(n-6)*(2*n-9)*(2*n-11)*a(n-6)=0. - R. J. Mathar, Jun 07 2016

Mathematica

CoefficientList[Series[(-1 + Sqrt[-3 + 4*Sqrt[1-4*x] + 8*x])/(-2 + 2*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)

Prog

(Maxima)
a(n):=sum(2^k*binomial(2*k+1, k)*binomial(2*n-2*k-2, n-1), k, 0, (n-1)/2)/n;
(PARI) my(x='x+O('x^50)); concat([0], Vec((sqrt(8*x+4*sqrt(1-4*x)-3)-1)/(2*sqrt(1-4*x)-2))) \\ G. C. Greubel, Jun 02 2017

Crossrefs

Cf. A000108.

Sequence in context: A149263 A149264 A264331 * A149265 A149266 A149267

Adjacent sequences: A243757 A243758 A243759 * A243761 A243762 A243763

Keyword

nonn

Author

Vladimir Kruchinin, Jun 10 2014

Status

approved