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%I #9 Jan 09 2018 14:30:01
%S 1,1,3,2,5,-2,3,-20,-2,-38,75,84,612,596,2293,534,4722,-6354,6716,
%T -27696,44962,-8232,391279,310072,1975046,906028,6363470,-2135522,
%U 15639296,-21729494,57269753,-46268934,370270484,130699742,2081100420,915348168,8523787094
%N G.f. satisfies: A(x) = 1 + x*A(x)/A(-x) + x^2*A(x)*A(-x).
%H Paul D. Hanna, <a href="/A208889/b208889.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 1 + x + 3*x^2 + 2*x^3 + 5*x^4 - 2*x^5 + 3*x^6 - 20*x^7 +...
%e Related series:
%e A(x)/A(-x) = 1 + 2*x + 2*x^2 - 2*x^4 - 12*x^5 - 20*x^6 - 38*x^7 - 38*x^8 +...
%e A(x)*A(-x) = 1 + 5*x^2 + 15*x^4 + 36*x^6 + 87*x^8 + 320*x^10 + 1567*x^12 +...
%t terms = 37; A[_] = 1; Do[A[x_] = 1 + x*A[x]/A[-x] + x^2*A[x]*A[-x] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 09 2018 *)
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*A/subst(A,x,-x+x*O(x^n))+x^2*A*subst(A,x,-x+x*O(x^n)));polcoeff(A,n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A209199.
%K sign
%O 0,3
%A _Paul D. Hanna_, Mar 09 2012
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