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%I #21 Feb 03 2023 09:05:24
%S 1,4,8,8,0,-8,0,16,0,-40,0,112,0,-336,0,1056,0,-3432,0,11440,0,-38896,
%T 0,134368,0,-470288,0,1664096,0,-5943200,0,21395520,0,-77558760,0,
%U 282861360,0,-1037158320,0,3821109600,0,-14138105520,0,52512963360,0,-195730136160
%N G.f. A(x) = 1 + 4*x*A(x)^(1/2); A(x) = 1 + 8*x^2 + 4*x*sqrt(1 + 4*x^2).
%F a(n) = -4^n*binomial(n/2,n)/(n/2 - 1), except a(2) = 8, for n>=0.
%F G.f.: (exp(asinh(2*x)))^2. - _Philippe Deléham_, Feb 01 2012
%F D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +4*(-n+4)*a(n-2) +4*(-n+5)*a(n-3)=0. - _R. J. Mathar_, Jan 23 2020
%F From _Alexander Burstein_, Mar 27 2022: (Start)
%F G.f. satisfies: A(-x) = 1/A(x).
%F a(2*n+3) = (-1)^n*8*A000108(n) for n>=0. (End)
%o (PARI) a(n)=4^n*if(n==2,1/2,binomial(n/2,n)/(1-n/2))
%Y Cf. A135864, A214377, A000108.
%K sign
%O 0,2
%A _Paul D. Hanna_, Dec 02 2007
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