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%I #6 Jun 04 2012 14:25:39
%S 1,1,2,8,26,127,478,2536,10250,56900,239880,1370272,5940054,34607146,
%T 153018932,904441648,4058644842,24254529036,110096276440,663665021280,
%U 3040205250984,18455364854839,85176971647470,520059936017128
%N G.f. satisfies: A(x) = 1 + x*A(x)^3*A(-x).
%e G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 127*x^5 + 478*x^6 +...
%e Compare bisections of A(x)^2, A(x)^2*A(-x), and A(x)^4*A(-x)^2:
%e A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 72*x^4 + 338*x^5 + 1378*x^6 + 6952*x^7 +...
%e A(x)^2*A(-x) = 1 + x + 5*x^2 + 11*x^3 + 72*x^4 + 191*x^5 + 1378*x^6 + 3979*x^7 +...
%e A(x)^4*A(-x)^2 = 1 + 2*x + 11*x^2 + 32*x^3 + 191*x^4 + 636*x^5 + 3979*x^6 +...
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 9*x^2 + 37*x^3 + 144*x^4 + 669*x^5 + 2882*x^6 + 14229*x^7 +...
%e A(x)^3*A(-x) = 1 + 2*x + 8*x^2 + 26*x^3 + 127*x^4 + 478*x^5 + 2536*x^6 +...
%e A(x)^3*A(-x)^2 = 1 + x + 8*x^2 + 14*x^3 + 127*x^4 + 264*x^5 + 2536*x^6 +...
%o (PARI) {a(n)=local(A=1+x+O(x^21));for(i=0,n,A=1+x*A^3*subst(A,x,-x));polcoeff(A,n)}
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 09 2008
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