A098333 Expansion of 1/sqrt(1 - 2x + 13x^2).

1, 1, -5, -17, 19, 211, 181, -2015, -5837, 12259, 91585, 29965, -1033955, -2347955, 7953115, 43864543, -11941037, -559875245, -942036911, 5060812717, 21502740649, -20676139991, -307241918945, -344022187613

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• easy
• sign

Comments

Central coefficients of (1 + x - 3x^2)^n.

Binomial transform of 1/sqrt(1+12x^2), or (1,0,-6,0,54,0,-540,...).

Binomial transform is A012000.

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

Crossrefs

Cf. A098331, A098332, A098334.

Programs

Maple

a := n -> hypergeom([1/2 - n/2, -n/2], [1], -12):
seq(simplify(a(n)), n=0..23); # Peter Luschny, Mar 19 2018

Mathematica

CoefficientList[Series[1/Sqrt[1-2*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 09 2014 *)

PARI

x='x+O('x^99); Vec(1/(1-2*x+13*x^2)^(1/2)) \\ Altug Alkan, Mar 18 2018

Formula

E.g.f.: exp(x)*BesselI(0, 2*sqrt(-3)x);

a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)(-3)^k;

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)(-3)^k.

D-finite with recurrence: n*a(n) + (-2*n+1)*a(n-1) + 13*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012

Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Feb 09 2014

a(n) = (sqrt(13))^n*P(n,1/sqrt(13)), where P(n,x) is the Legendre polynomial of degree n. - Peter Bala, Mar 18 2018

a(n) = hypergeom([1/2 - n/2, -n/2], [1], -12). - Peter Luschny, Mar 19 2018