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%I #7 Jan 27 2013 12:06:11
%S 1,1,1,2,5,15,97,528,7525,81413,2330257,53229494,2883887781,
%T 143434131379,14268520503521,1569684574492244,282300076226827269,
%U 69232924049740233209,22337274702237239579729,12261557957001826246975754,7069170473480519272781373829
%N G.f.: A(x) = Sum_{n>=0} x^n*(A(n*x) + A(-n*x))/2.
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 97*x^6 + 528*x^7 +...
%e where
%e A(x) = 1 + x*(A(x)+A(-x))/2 + x^2*(A(2*x)+A(-2*x))/2 + x^3*(A(3*x)+A(-3*x))/2 + x^4*(A(4*x)+A(-4*x))/2 +...
%e Related expansions begin:
%e (A(x)+A(-x))/2 = 1 + x^2 + 5*x^4 + 97*x^6 + 7525*x^8 +...
%e (A(2*x)+A(-2*x))/2 = 1 + 4*x^2 + 80*x^4 + 6208*x^6 +...
%e (A(3*x)+A(-3*x))/2 = 1 + 9*x^2 + 405*x^4 + 70713*x^6 +...
%e (A(4*x)+A(-4*x))/2 = 1 + 16*x^2 + 1280*x^4 + 397312*x^6 +...
%e (A(5*x)+A(-5*x))/2 = 1 + 25*x^2 + 3125*x^4 + 1515625*x^6 +...
%e (A(6*x)+A(-6*x))/2 = 1 + 36*x^2 + 6480*x^4 + 4525632*x^6 +...
%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(k=1, n, A=1+sum(j=1, n, x^j*(subst(A, x, j*x)+subst(A, x,-j*x))/2)); polcoeff(A, n)}
%Y Cf. A196195, A210525.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Sep 29 2011
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