A212205 - OEIS

%I #16 Jan 30 2020 21:29:17

%S 1,1,2,4,8,18,36,86,172,426,852,2162,4324,11166,22332,58438,116876,

%T 309042,618084,1648154,3296308,8851206,17702412,47813790,95627580,

%U 259585002,519170004,1415431266,2830862532,7747200558,15494401116,42545600310,85091200620,234346445154,468692890308,1294260644906,2588521289812,7165245015510,14330490031020

%N G.f.: ((1+2*x)*sqrt(1-6*x^2+x^4)-1+5*x^2-2*x^3)/(2*x*(1-6*x^2)).

%H Alois P. Heinz, <a href="/A212205/b212205.txt">Table of n, a(n) for n = 0..1000</a>

%H D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i2p33">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.

%F a(n) ~ 2*2^(3/4)*(1+sqrt(2))^(n+3)/(n^(3/2)*sqrt(Pi)) if n is even and a(n) ~ 2^(3/4)*(1+sqrt(2))^(n+4)/(n^(3/2)*sqrt(Pi)) if n is odd. - _Vaclav Kotesovec_, May 21 2013

%F Conjecture D-finite with recurrence: (n+1)*a(n) +2*(n-1)*a(n-1) +6*(-2*n+1)*a(n-2) +12*(-2*n+5)*a(n-3) +(37*n-77)*a(n-4) +2*(37*n-151)*a(n-5) +6*(-n+5)*a(n-6) +12*(-n+7)*a(n-7)=0. - _R. J. Mathar_, Aug 20 2018

%p a:= proc(n) option remember; `if`(n<6, [1, 1, 2, 4, 8, 18][n+1],

%p       (-132*a(n-1) +(660-834*n+84*n^2)*a(n-2) +804*a(n-3)

%p       +(2981*n-6690-259*n^2)*a(n-4) -72*a(n-5) +6*(n-6)*(7*n-59)*a(n-6))

%p       / ((n+1)*(7*n-66)))

%p     end:

%p seq(a(n), n=0..40); # _Alois P. Heinz_, May 21 2013

%t CoefficientList[Series[((1+2*x)*Sqrt[1-6*x^2+x^4]-1+5*x^2-2*x^3)/(2*x*(1-6*x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, May 21 2013 *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, May 11 2012