A098335 Expansion of 1/sqrt(1-4x+8x^2).

1, 2, 2, -4, -26, -68, -76, 184, 1222, 3308, 3772, -9656, -64676, -177448, -203992, 536176, 3607622, 9968972, 11510636, -30723416, -207302156, -575382392, -666187432, 1796105744, 12142184476, 33803271032

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• easy
• sign

Comments

Central coefficients of (1+2x-x^2)^n. Binomial transform of A098331.

Diagonal of rational function 1/(1 - (x^2 + 2*x*y - y^2)). - Gheorghe Coserea, Aug 04 2018

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Programs

Mathematica

a[n_] := 2^n*Hypergeometric2F1[1/2 - n/2, -n/2, 1, -1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 05 2012, after Alexander R. Povolotsky *)
CoefficientList[Series[1/Sqrt[1 - 4*x + 8*x^2], {x,0,50}], x] (* G. C. Greubel, Feb 19 2017 *)

PARI

x='x+O('x^25); Vec(1/sqrt(1 - 4*x + 8*x^2)) \\ G. C. Greubel, Feb 19 2017

Formula

From Paul Barry, Sep 08 2004: (Start)

E.g.f. : exp(2*x)*BesselI(0, 2*I*x), I=sqrt(-1);

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*binomial(n-k,k)*2^n*(-4)^(-k).

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

E.g.f. : exp(2*x)*BesselJ(0, 2*x). - Sergei N. Gladkovskii, Aug 22 2012

It appears that a(j+2) = (2*(2*j+1)*a(j+1))/(j+1)-(8*j*a(j))/(j+1), in case of re-indexing from 0 to 1. - Alexander R. Povolotsky, Aug 22 2012

D-finite with recurrence: a(n+2) = ((4*n+6)*a(n+1) - 8*(n+1)*a(n))/(n+2); a(0)=1,a(1)=2. - Sergei N. Gladkovskii, Aug 22 2012

a(n) = 2^n*2F_1(1/2-n/2, -n/2, 1, -1), where _2F_1(a,b;c;x) is the hypergeometric function. - _Alexander R. Povolotsky, Aug 22 2012