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A212922 G.f. satisfies: A(x) = x^2/(1-x) + Series_Reversion(x - x*A(x)).

%I A212922 #7 Aug 24 2017 10:48:24
%S A212922 1,2,5,21,120,800,5881,46565,391876,3473879,32226510,311313683,
%T A212922 3119693862,32333294383,345754479372,3807294710182,43101806735623,
%U A212922 500977869387150,5971566838065819,72925079326977943,911614856156206061,11656341547670071145,152347288068103795503
%N A212922 G.f. satisfies: A(x) = x^2/(1-x) + Series_Reversion(x - x*A(x)).
%H A212922 Vaclav Kotesovec, <a href="/A212922/b212922.txt">Table of n, a(n) for n = 1..270</a>
%F A212922 G.f. A(x) also satisfies:
%F A212922 (1) A(x) = x/(1-x) + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
%F A212922 (2) A(x) = x^2/(1-x) + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).
%e A212922 G.f.: A(x) = x + 2*x^2 + 5*x^3 + 21*x^4 + 120*x^5 + 800*x^6 + 5881*x^7 +...
%e A212922 The series reversion of x - x*A(x) begins:
%e A212922 x + x^2 + 4*x^3 + 20*x^4 + 119*x^5 + 799*x^6 + 5880*x^7 +...
%e A212922 which equals A(x) - x^2/(1-x).
%e A212922 The g.f. A(x) satisfies:
%e A212922 A(x) - x^2/(1-x) = 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 A212922 log(A(x)/x - x/(1-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 A212922 (PARI) {a(n)=local(A=x+x^2); for(i=1, n, A=x^2/(1-x+x*O(x^n))+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
%o A212922 for(n=1, 25, print1(a(n), ", "))
%o A212922 (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o A212922 {a(n)=local(A=x+x^2); for(i=1, n, A=x/(1-x+x*O(x^n))+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
%o A212922 for(n=1, 25, print1(a(n), ", "))
%o A212922 (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o A212922 {a(n)=local(A=x+x^2); for(i=1, n, A=x^2/(1-x+x*O(x^n))+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
%o A212922 for(n=1, 25, print1(a(n), ", "))
%Y A212922 Cf. A212923, A088714, A212910, A212919.
%K A212922 nonn
%O A212922 1,2
%A A212922 _Paul D. Hanna_, May 31 2012