A098338 Expansion of 1/sqrt(1-6x+13x^2).

1, 3, 7, 9, -21, -207, -911, -2769, -5213, 2457, 74997, 400491, 1409109, 3323583, 2219343, -27453951, -186624333, -750905127, -2088947819, -2955863589, 8506703569, 86421384387, 401183114163, 1280139325101, 2522745571021

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• easy
• sign

Comments

Binomial transform of A098335. Second binomial transform of A098331.

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

Links

• Robert Israel, Table of n, a(n) for n = 0..1796
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• Robert Israel, Plot of a(n) sqrt(n)/13^(n/2) for 1<=n<=10000.
• Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Programs

Maple

f:= gfun:-rectoproc({(13*n+13)*a(n)+(-9-6*n)*a(n+1)+(n+2)*a(n+2), a(0)=1, a(1)=3},a(n),remember):
map(f, [$0..50]); # Robert Israel, Mar 02 2017

Mathematica

CoefficientList[Series[1/Sqrt[1-6*x+13*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)

Formula

E.g.f.: exp(3*x)*BesselI(0, 2*I*x), I=sqrt(-1).

a(n) = Sum{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k)*3^n*(-9)^(-k).

a(n) = Sum{k=0..floor(n/2)} binomial(n, 2k)*binomial(2k, k)*3^n*(-9)^(-k).

D-finite with recurrence: n*a(n) +3*(1-2*n)*a(n-1) +13*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012

Recurrence follows from the differential equation (13x-3) g(x) + (13x^2-6x+1) g'(x) = 0 satisfied by the generating function. - Robert Israel, Mar 02 2017

Lim sup n->infinity |a(n)|^(1/n) = sqrt(13). - Vaclav Kotesovec, Sep 29 2013