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%I #36 Oct 01 2019 09:12:55
%S 1,1,2,8,32,178,944,6255,39366,293652,2090576,17085798,134136792,
%T 1182991528,10085875720,95087538324,871536657504,8727880568468,
%U 85385942061016,904071273001352,9389429908430784,104728235042891360,1149676904405092704,13467595558130095308,155705728677310569008
%N G.f. satisfies: A(x + A(x)*A(-x)) = x.
%H Paul D. Hanna, <a href="/A141202/b141202.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) A(x) = x - A(-A(x)) * A(A(x)).
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / n!.
%F (3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / (n!*x) ).
%F (4) A(-I*x) * A(I*x) = F(x), where F(x) is the g.f. of A263530 and satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1.
%e G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 +...
%e By definition, Series_Reversion(A(x)) = x + A(-x)*A(x) where
%e A(-x)*A(x) = -x^2 - 3*x^4 - 52*x^6 - 1596*x^8 - 68174*x^10 - 3679964*x^12 +...+ (-1)^n * A263530(n)*x^(2*n) +...
%e Consequently, A(x) = x - A(-A(x))*A(A(x)) where
%e -A(-A(x)) = x + 0*x^2 + 2*x^3 + x^4 + 30*x^5 + 38*x^6 + 852*x^7 +...
%e A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 134*x^5 + 786*x^6 + 4852*x^7 +...
%e The related g.f. of A263530, F(x) = A(-I*x)*A(I*x), satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
%e F(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 +...
%t m = 26; A[_] = 0;
%t Do[A[x_] = x - A[-A[x]] A[A[x]] + O[x]^m // Normal, {m}];
%t CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Oct 01 2019 *)
%o (PARI) {a(n)=local(A=x+x^2);for(i=0,n,A=serreverse(x+A*subst(A,x,-x+x*O(x^n)))) ;polcoeff(A,n)}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (A*subst(-A, x, -x))^m/m!))+x*O(x^n)); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A263530, A227852.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 13 2008, Sep 05 2008
%E Edited by _N. J. A. Sloane_, Sep 13 2008 at the suggestion of R. J. Mathar
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