A108476 Expansion of (1-4*x)/(1-6*x-12*x^2+8*x^3).

1, 2, 24, 160, 1232, 9120, 68224, 508928, 3799296, 28357120, 211662848, 1579868160, 11792306176, 88018952192, 656982441984, 4903783628800, 36602339459072, 273203580764160, 2039219289063424, 15220939987877888

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Comments

In general, Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*r^j has expansion (1 - (r+1)*x)/(1 - (r+3)*x - (r-1)*(r+3)*x^2 + (r-1)^3*x^3).

Links

• Index entries for linear recurrences with constant coefficients, signature (6,12,-8).

Programs

Mathematica

CoefficientList[Series[(1-4x)/(1-6x-12x^2+8x^3),{x,0,30}],x] (* or *) LinearRecurrence[{6,12,-8},{1,2,24},30] (* Harvey P. Dale, Feb 21 2012 *)

Formula

G.f.: (1-4*x)/((1+2*x)*(1-8*x+4*x^2)).

a(n) = 6*a(n-1)+12*a(n-2)-8*a(n-3).

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(2*(n-k), j)*C(2*k, j)*3^j.

Conjecture: a(n) = A002605(n+1)*A026150(n). - R. J. Mathar, Jul 08 2009

a(0)=1, a(1)=2, a(2)=24, a(n)=6*a(n-1)+12*a(n-2)-8*a(n-3). - Harvey P. Dale, Feb 21 2012

a(n) = (-2)^n/2 +A102591(n)/2. - R. J. Mathar, Sep 20 2012