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%I #23 Nov 06 2019 04:34:52
%S 1,-1,4,-24,178,-1512,14152,-142705,1528212,-17211564,202460400,
%T -2474708496,31310415376,-408815254832,5495451727376,-75907303147652,
%U 1075685334980240,-15618612118252960,232102241507321384,-3526880759915999016
%N G.f. satisfies: x = A( x + A(x)^2 ).
%C Signed version of A213591.
%H Paul D. Hanna, <a href="/A139702/b139702.txt">Table of n, a(n), n=1..100.</a>
%F Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then G(x) = G(G(x)) - x^2 = g.f. of A138740.
%F G.f. satisfies: A(x) = x*G(-A(x)^2/x) where G(x) = 1 + x*G(1-1/G(x))^2 is the g.f. of A212411.
%F G.f.: A(x)/x is the unique solution to variable A in the infinite system of simultaneous equations starting with:
%F A = 1 - x*B^2;
%F B = A - x*C^2;
%F C = B - x*D^2;
%F D = C - x*E^2;
%F E = D - x*F^2; ...
%F G.f. satisfies: A(x) = x*exp( Sum_{n>=0} (-1)^(n+1)*[d^n/dx^n A(x)^(2n+2)/x]/(n+1)! ). [_Paul D. Hanna_, Dec 18 2010]
%e G.f.: A(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
%e A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
%e where A(x + A(x)^2) = x.
%e Let G(x) = Series_Reversion( A(x) ) = x + A(x)^2, then:
%e G(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+... and
%e G(G(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 -+...
%e so that G(x) = G(G(x)) - x^2 = g.f. of A138740.
%e Logarithmic series:
%e log(A(x)/x) = -A(x)^2/x + [d/dx A(x)^4/x]/2! - [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! -+...
%t nmax = 20; sol = {a[1] -> 1}; nmin = Length[sol]+1;
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[x - A[x + A[x]^2] + O[x]^(n+1), x][[nmin;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nmin, nmax}];
%t a /@ Range[nmax] /. sol (* _Jean-François Alcover_, Nov 06 2019 *)
%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(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_, Dec 18 2010] */
%o {a(n)=local(A=x-x^2+x*O(x^n));for(i=1,n,
%o A=x*exp(sum(m=0,n,(-1)^(m+1)*Dx(m,A^(2*m+2)/x)/(m+1)!)+x*O(x^n)));polcoeff(A,n)}
%Y Cf. A138740, A212411, A213591.
%Y Cf. A088714, A088717, A091713, A120971, A140094, A140095.
%Y Cf. A143426, A087949, A143435, A182969.
%K sign
%O 1,3
%A _Paul D. Hanna_, Apr 30 2008, May 20 2008
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