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 A141372 G.f. satisfies: A(x) = x + A(A(A(x)))^2. 4

%I

%S 1,1,6,57,684,9512,146848,2455208,43764802,822963750,16203122280,

%T 332189276516,7062047285812,155178233311932,3515420453148936,

%U 81936668615592785,1961578144170589430,48167700575393576406

%N G.f. satisfies: A(x) = x + A(A(A(x)))^2.

%F G.f. A(x) satisfies:

%F (1) A( x - A(A(x))^2 ) = x.

%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x))^(2*n) / n!.

%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(A(x))^(2*n)/x / n! ).

%e G.f.: A(x) = x + x^2 + 6*x^3 + 57*x^4 + 684*x^5 + 9512*x^6 +...

%e The g.f. satisfies the series:

%e A(x) = x + A(A(x))^2 + d/dx A(A(x))^4/2! + d^2/dx^2 A(A(x))^6/3! + d^3/dx^3 A(A(x))^8/4! +...

%e as well as the logarithmic series:

%e log(A(x)/x) = A(A(x))^2/x + [d/dx A(A(x))^4/x]/2! + [d^2/dx^2 A(A(x))^6/x]/3! + [d^3/dx^3 A(A(x))^8/x]/4! +...

%e Related expansions.

%e A(A(x)) = x + 2*x^2 + 14*x^3 + 145*x^4 + 1848*x^5 + 26920*x^6 +...

%e A(A(A(x))) = x + 3*x^2 + 24*x^3 + 270*x^4 + 3658*x^5 + 55970*x^6 +...

%e A(A(A(x)))^2 = x^2 + 6*x^3 + 57*x^4 + 684*x^5 + 9512*x^6 +...

%e The series reversion of A(x) = x - A(A(x))^2, where

%e A(A(x))^2 = x^2 + 4*x^3 + 32*x^4 + 346*x^5 + 4472*x^6 + 65292*x^7 +...

%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A^2,x,subst(A,x,A+x*O(x^n))));polcoeff(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, subst(A,x,A+x*O(x^n))^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}

%o for(n=1,25,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*exp(sum(m=1, n, Dx(m-1, subst(A,x,A+x*O(x^n))^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A141370, A141371.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jun 28 2008

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Last modified February 27 04:56 EST 2020. Contains 332299 sequences. (Running on oeis4.)