A098456 Expansion of 1/sqrt(1-4*x-64*x^2).

1, 2, 38, 212, 2566, 20092, 210524, 1884136, 18854854, 178415852, 1764019828, 17115907096, 169100140444, 1661540282456, 16458178007288, 162887627833552, 1618680238292294, 16095872154638156, 160435286115927044, 1600771362880092472, 15997473711080724916

Offset: 0

Author

Paul Barry, Sep 08 2004

Keywords

• easy
• nonn
• changed

Comments

Define Q(n,x)=sum{k=0..floor(n/2), binomial(n,k)binomial(2(n-k),n)x^(n-2k)}. Then a(n)=4^n*Q(n,1/4). Central coefficients of (1+2*x+17*x^2)^n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Crossrefs

Cf. A084770, A098455.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-4x-64x^2],{x,0,30}],x] (* Harvey P. Dale, Dec 27 2011 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-4*x-64*x^2)) \\ Joerg Arndt, May 11 2013

Formula

E.g.f.: exp(2x) * BesselI(0, 2*sqrt(17)*x).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), n)*16^k.

D-finite with recurrence: n*a(n) +2*(1-2*n)*a(n-1) +64*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012

a(n) ~ sqrt(578+34*sqrt(17))*(2+2*sqrt(17))^n/(34*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 15 2012

From Seiichi Manyama, Aug 30 2025: (Start)

a(n) = Sum_{k=0..n} (1-4*i)^k * (1+4*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.

a(n) = Sum_{k=0..floor(n/2)} 17^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). (End)