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%I #11 Sep 26 2017 07:36:15
%S 1,1,-2,9,-56,420,-3572,33328,-334354,3559310,-39838760,465743720,
%T -5658983108,71191948512,-924554859776,12365546196641,
%U -169995491295312,2398380272232272,-34680290150700800,513390937937217088,-7773229533145403728
%N G.f. satisfies A(x) = A(A(x)) - x^2 with A(0)=0.
%F G.f. satisfies: A(x) = x + x^2*exp( 2*Sum_{n>=0} [d^n/dx^n (x-A(x))^(n+1)/x]/(n+1)! ). - _Paul D. Hanna_, Mar 24 2011
%F Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then G(x)^2 = A(x) - x so that G(x + G(x)^2) = x.
%e G.f.: A(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-..;
%e A(A(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
%e The g.f. satisfies:
%e A(x) = x + x^2*exp((x-A(x))/x + [d/dx (x-A(x))^2/x]/2! + [d^2/dx^2 (x-A(x))^3/x]/3! + [d^3/dx^3 (x-A(x))^4/x]/4! +...)^2.
%e Higher order iterations of A=A(x) may be expressed in terms of A and x:
%e A(A(x)) = A + x^2 ;
%e A(A(A(x))) = (A + A^2) + x^2 ;
%e A(A(A(A(x)))) = (A + 2*A^2) + (1 + 2*A)*x^2 + x^4 ;
%e A(A(A(A(A(x))))) = (A + 3*A^2 + 2*A^3 + A^4) + (1 + 4*A + 2*A^2)*x^2 + 2*x^4 ;
%e A(A(A(A(A(A(x)))))) = (A + 4*A^2 + 6*A^3 + 5*A^4) + (1 + 6*A + 10*A^2 + 8*A^3)*x^2 + (3 + 6*A + 8*A^2)*x^4 + (2 + 4*A)*x^6 + x^8 ;
%e A(A(A(A(A(A(A(x))))))) = (A + 5*A^2 + 12*A^3 + 18*A^4 + 14*A^5 + 10*A^6 + 4*A^7 + A^8) + (1 + 8*A + 24*A^2 + 40*A^3 + 30*A^4 + 16*A^5 + 4*A^6)*x^2 + (4 + 18*A + 40*A^2 + 24*A^3 + 8*A^4)*x^4 + (6 + 20*A + 8*A^2)*x^6 + 5*x^8 .
%e The sums of coefficients in the above expansions form A000278: [1,1,2,3,7,16,65,321,4546,107587,20773703,...].
%e Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then
%e G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
%e G(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
%e so that G(x)^2 = A(x) - x and G(x + G(x)^2) = x.
%o (PARI) {a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n,A=A-polcoeff(subst(A,x,A+x*O(x^i))-x^2,i)*x^i);polcoeff(A,n))}
%o (PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(serreverse(A), n))}
%o (PARI) /* n-th Derivative: */
%o {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o /* G.f.: [Paul D. Hanna, Mar 24 2011] */
%o {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+x^2*exp(2*sum(m=0, n, Dx(m, (x-A)^(m+1)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
%Y Cf. A000278, A138739.
%Y Cf. A139702, A276370.
%K sign
%O 1,3
%A _Paul D. Hanna_, Mar 26 2008, Mar 27 2008, Apr 30 2008
%E Edited by _Paul D. Hanna_, May 16 2010
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