A098334 Expansion of 1/sqrt(1-2x+17x^2).

1, 1, -7, -23, 49, 401, 41, -5767, -11423, 65569, 299353, -441847, -5511791, -3665999, 79937417, 212712857, -861871423, -5076450239, 3966949049, 89482678313, 110424995569, -1233175514671, -4202194115863, 11830822055353, 91629438996001, -13485315511199

Offset

0,3

Comments

Central coefficients of (1+x-4x^2)^n.

Binomial transform of 1/sqrt(1+16x^2), or (1,0,-8,0,96,0,-1280,...).

Binomial transform is A098337.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Formula

E.g.f.: exp(x)*BesselI(0, 4*I*x), I=sqrt(-1);

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

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

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

D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +17*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012

Lim sup n->oo |a(n)|^(1/n) = sqrt(17). - Vaclav Kotesovec, Feb 09 2014

a(n) = hypergeom([-n/2, 1/2-n/2], [1], -16). - Peter Luschny, Sep 18 2014

a(n) = (sqrt(17))^n*P(n,1/sqrt(17)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018

Maple

a := n -> hypergeom([-n/2, 1/2-n/2], [1], -16);
seq(round(evalf(a(n), 99)), n=0..28); # Peter Luschny, Sep 18 2014

Mathematica

CoefficientList[Series[1/Sqrt[1-2*x+17*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)

Prog

(PARI) x='x+O('x^99); Vec(1/(1-2*x+17*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

Crossrefs

Cf. A098331, A098332, A098333.

Sequence in context: A180044 A062725 A147121 * A299255 A038796 A332492

Adjacent sequences: A098331 A098332 A098333 * A098335 A098336 A098337

Keyword

easy,sign

Author

Paul Barry, Sep 03 2004

Status

approved