A098336 Expansion of 1/sqrt(1 - 4*x + 12*x^2).

1, 2, 0, -16, -56, -48, 384, 1920, 3168, -8512, -66560, -161280, 113920, 2224640, 7311360, 3354624, -69253632, -306754560, -408059904, 1898029056, 12054196224, 25377005568, -38874316800, -443400781824, -1289598418944

Offset: 0

Author

Paul Barry, Sep 03 2004

Keywords

• sign
• easy

Comments

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

Binomial transform of A098332.

Diagonal of rational function 1/(1 - (2*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
• 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.

Programs

Maple

A098336 := n -> 2^n*hypergeom([-n/2, 1/2-n/2], [1], -2);
seq(round(evalf(A098336(n),99)),n=0..30); # Peter Luschny, Sep 18 2014

Mathematica

CoefficientList[Series[1/Sqrt[1-4*x+12*x^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 09 2013 *)

PARI

Vec(1/sqrt(1-4*x+12*x^2) + O(x^50)) \\ G. C. Greubel, Jan 30 2017

Formula

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

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(2(n-k), k)*(-3)^k. - Paul Barry, Sep 08 2004

D-finite with recurrence: n*a(n) + 2*(-2*n+1)*a(n-1) + 12*(n-1)*a(n-2) = 0. - R. J. Mathar, Nov 24 2012

G.f.: G(0), where G(k) = 1 + 2*x*(1-3*x)*(4*k+1)/( 2*k+1 - x*(1-3*x)*(2*k+1)*(4*k+3)/(x*(1-3*x)*(4*k+3) + (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 30 2013

Lim sup_{n->infinity} |a(n)|/((2*sqrt(3))^n/sqrt(Pi*n)) = 6^(1/4). - Vaclav Kotesovec, Oct 09 2013

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