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Expansion of (1/(1-2*x^2))*c(x/(1-2*x^2)), where c(x) is the g.f. of A000108.
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%I #19 Jan 09 2023 14:49:43

%S 1,1,4,9,30,94,328,1165,4294,16134,61752,239610,940716,3729324,

%T 14908176,60026109,243211206,990897478,4057013080,16683617326,

%U 68879236036,285388549188,1186296731376,4945790840338,20675513743900,86648395759516

%N Expansion of (1/(1-2*x^2))*c(x/(1-2*x^2)), where c(x) is the g.f. of A000108.

%H Vincenzo Librandi, <a href="/A105865/b105865.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: (1-sqrt((1-4*x-2*x^2)/(1-2*x^2)))/(2*x).

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

%F Conjecture: (n+1)*a(n) +2(1-2n)*a(n-1) +4*(1-n)*a(n-2) +4*(2n-3)*a(n-3) +4*(n-3)*a(n-4)=0. - _R. J. Mathar_, Dec 13 2011

%F a(n) ~ 3^(1/4) * (2+sqrt(6))^(n+1) / (2^(9/4) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 01 2014

%F G.f.: 1/G(x), with G(x) = 1-2*x^2-(x-x^3)/(1-x^2-(x-x^3)/G(x)) (continued fraction). - _Nikolaos Pantelidis_, Jan 09 2023

%t CoefficientList[Series[(1-Sqrt[(1-4*x-2*x^2)/(1-2*x^2)])/(2*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%o (PARI) x='x+O('x^50); Vec((1-sqrt((1-4*x-2*x^2)/(1-2*x^2)))/(2*x)) \\ _G. C. Greubel_, Mar 16 2017

%K easy,nonn

%O 0,3

%A _Paul Barry_, Apr 23 2005