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%I #3 Jun 30 2015 23:31:18
%S 1,1,6,58,720,10506,172284,3092717,59758608,1228626514,26657057728,
%T 606616602302,14410894287172,356081682054300,9124705519233832,
%U 241916247567814448,6622686675121529288,186900262172114801748,5429779249015331564288,162190080378495122207760
%N G.f. A(x) satisfies: A(x)*A'(x) = Series_Reversion( x - 2*x*A(x)*A'(x) - A(x)^2 ).
%C Self-convolution yields A259611.
%e G.f. A(x) = x + x^2 + 6*x^3 + 58*x^4 + 720*x^5 + 10506*x^6 + 172284*x^7 +...
%e where
%e A(x)*A'(x) = x + 3*x^2 + 26*x^3 + 320*x^4 + 4776*x^5 + 81018*x^6 + 1510336*x^7 + 30328173*x^8 +...+ A259610(n)*x^n +...
%e also
%e A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 128*x^5 + 1592*x^6 + 23148*x^7 + 377584*x^8 + 6739594*x^9 +...+ A259611(n)*x^n +...
%o (PARI) {a(n)=local(A=x^2); for(i=0, n, A = 2*intformal( serreverse(x - x*A' - A +x*O(x^n)))); polcoeff(sqrt(A), n)}
%o for(n=1, 25, print1(a(n), ", "))
%Y Cf. A259610, A259611.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 30 2015