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A212923 G.f. satisfies: A(x) = x^2 + Series_Reversion(x - x*A(x)).

%I A212923 #10 Feb 21 2014 05:48:02
%S A212923 1,2,4,19,111,734,5338,41839,348827,3065255,28199803,270253498,
%T A212923 2687629926,27652068276,293627150268,3211604669731,36124424800797,
%U A212923 417294625090201,4944772338009206,60045368928594948,746560751627818906,9496624640844863631,123507266690219103213
%N A212923 G.f. satisfies: A(x) = x^2 + Series_Reversion(x - x*A(x)).
%C A212923 This is an application of the more general formula given by:
%C A212923 if G(x) = Series_Reversion(x - x*F(x)), with F(0)=0, then
%C A212923 (1) G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*F(x)^n/n!,
%C A212923 (2) G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*F(x)^n/n! );
%C A212923 here F(x) = A(x) and G(x) = A(x) - x^2.
%H A212923 Vaclav Kotesovec, <a href="/A212923/b212923.txt">Table of n, a(n) for n = 1..365</a>
%F A212923 G.f. A(x) also satisfies:
%F A212923 (1) A(x) = x+x^2 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
%F A212923 (2) A(x) = x^2 + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).
%e A212923 G.f.: A(x) = x + 2*x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
%e A212923 The series reversion of x - x*A(x) begins:
%e A212923 x + x^2 + 4*x^3 + 19*x^4 + 111*x^5 + 734*x^6 + 5338*x^7 +...
%e A212923 which equals A(x) - x^2.
%e A212923 The g.f. A(x) satisfies:
%e A212923 A(x) - x^2 = x + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
%e A212923 log(A(x)/x - x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
%o A212923 (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x^2+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
%o A212923 for(n=1, 25, print1(a(n), ", "))
%o A212923 (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o A212923 {a(n)=local(A=x+x^2); for(i=1, n, A=x+x^2+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
%o A212923 for(n=1, 25, print1(a(n), ", "))
%o A212923 (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o A212923 {a(n)=local(A=x+x^2); for(i=1, n, A=x^2+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
%o A212923 for(n=1, 25, print1(a(n), ", "))
%Y A212923 Cf. A212922, A088714, A212910, A212919.
%K A212923 nonn
%O A212923 1,2
%A A212923 _Paul D. Hanna_, May 31 2012