A098106 - OEIS

%I #20 Nov 16 2022 09:45:12

%S 1,3,6,-8,-72,-144,64,960,1920,-512,-10752,-21504,4096,110592,221184,

%T -32768,-1081344,-2162688,262144,10223616,20447232,-2097152,-94371840,

%U -188743680,16777216,855638016,1711276032,-134217728,-7650410496,-15300820992,1073741824,67645734912,135291469824

%N Hankel transform of sequence (b(n)) where b(n) = Sum_{i=0..n} binomial(2*i,i).

%C Barker's g.f. is a simplification of the sum of the three g.f.s for a(3*n), a(3*n+1) and a(3*n+2): +(-1/(-8*x^3-1)) + (-3*x*(8*x^3-1)/(2*x+1)^2/(4*x^2-2*x+1)^2) + (-(-48*x^5+6*x^2)/(-64*x^6-16*x^3-1)). - _Georg Fischer_, Nov 16 2022

%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%F a(3*n) = (-8)^n; a(3*n+1) = 3*(-8)^n*(2*n+1); a(3*n+2) = 6*(-8)^n*(2*n+1).

%F G.f.: -(12*x^3-6*x^2+x-1) / (4*x^2-2*x+1)^2. - _Colin Barker_, Jun 27 2013

%F D-finite with recurrence: (2*n-3)^2*a(n) - (8*(n-2)^2+16*n-18)*a(n-1) + 4*(2*n-1)^2*a(n-2) = 0. - _Georg Fischer_, Nov 16 2022

%t CoefficientList[Series[-(12*x^3-6*x^2+x-1) / (4*x^2-2*x+1)^2, {x,0,35}], x] (* _Georg Fischer_, Nov 16 2022 *)

%o (PARI) a(n)=if(n<0,0,if(n%3,if(n%3-1,6*(-8)^floor(n/3)*(2*floor(n/3)+1),3*(-8)^floor(n/3)*(2*floor(n/3)+1)),(-8)^(n/3)))

%Y Cf. A006134.

%K sign,easy

%O 0,2

%A _Benoit Cloitre_, Sep 22 2004