|
%I #19 Jan 30 2020 21:29:17
%S 1,1,8,22,141,561,3291,15583,88691,459187,2599570,14136200,80391235,
%T 450046143,2579291352,14710321998,85002979083,491050703739,
%U 2859262171872,16674374605722,97747766045679,574231140306699,3385974360904227,20009363692187115,118582649963026677
%N Expansion of 2/(1-x+sqrt(1-2*x-27*x^2)).
%H Vincenzo Librandi, <a href="/A217275/b217275.txt">Table of n, a(n) for n = 0..200</a>
%F Generally for G.f. = 2/(1-x+sqrt(1-2x-(4*z-1)*x^2)) is asymptotic
%F a(n) ~ (1+2*sqrt(z))^(n+3/2)/(2*sqrt(Pi)*z^(3/4)*n^(3/2)); here we have the case z=7.
%F D-finite with recurrence: (n+2)*a(n)=(2*n+1)*a(n-1)+(4*z-1)*(n-1)*a(n-2);; here with z=7.
%F G.f.: 1/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - x - 7*x^2/(1 - ....))))), a continued fraction. - _Ilya Gutkovskiy_, May 26 2017
%t Table[SeriesCoefficient[2/(1-x+Sqrt[1-2*x-27*x^2]),{x,0,n}],{n,0,25}]
%t Table[Sum[Binomial[n,2k]*Binomial[2k,k]*7^k/(k+1),{k,0,n}],{n,0,25}]
%Y Cf. A001006 (z=1), A025235 (z=2), A025237 (z=3), A091147 (z=4), A091148 (z=5), A091149 (z=6).
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Sep 29 2012
|
