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%I #9 Mar 30 2012 18:36:46
%S 1,1,0,1,-3,13,-63,339,-1982,12429,-82827,582589,-4303016,33240205,
%T -267697961,2241725581,-19477340744,175259713769,-1630583565434,
%U 15663877511863,-155168272246709,1583282220672515,-16623104947488348,179409709469784087,-1988706708427161585
%N G.f. satisfies: A(A(x)) = A(x)/(1-x), so that the self-COMPOSE transform generates partial sums (A107098).
%F G.f. satisfies: A(x) = x + A(x)*Series_Reversion(A(x)).
%F Given g.f. A(x), let G(x) = Series_Reversion(A(x)), then G(x) satisfies:
%F (1) G(x) = 1 - x/A(x),
%F (2) G(x) = x - x*G(G(x)),
%F (3) -G(-x) is the g.f. of A030266, which shifts left under self-COMPOSE.
%e G.f.: A(x) = x + x^2 + x^4 - 3*x^5 + 13*x^6 - 63*x^7 + 339*x^8 -+...
%e If G(x) = series reversion of g.f. A(x) so that A(G(x)) = x, then G(x) begins:
%e G(x) = x - x^2 + 2*x^3 - 6*x^4 + 23*x^5 - 104*x^6 + 531*x^7 - 2982*x^8 -+...
%e Compare the functional inverse, G(x), to the arithmetic inverse x/A(x):
%e x/A(x) = 1 - x + x^2 - 2*x^3 + 6*x^4 - 23*x^5 + 104*x^6 - 531*x^7 + 2982*x^8 -+...
%o (PARI) {a(n)=local(A,B,F);if(n<1,0,F=x+2*x^2-3*x^3+x*O(x^n);A=F; for(j=0,n, for(i=0,j,B=serreverse(A);A=(A+subst(B,x, A/(1-x)))/2); A=round(A));polcoeff(A,n,x))}
%o (PARI) /* A(x) = x + A(x)*Series_Reversion(A(x)): */
%o {a(n)=local(A=x+x^2);for(i=1,n,A=x+A*serreverse(A+x*O(x^n)));polcoeff(A,n)}
%Y Cf. A107098.
%K sign
%O 1,5
%A _Paul D. Hanna_, May 12 2005, Jul 23 2011
%E Initial zero removed and offset changed to 1 by _Paul D. Hanna_, Jul 23 2011
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