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A138740 G.f. satisfies A(x) = A(A(x)) - x^2 with A(0)=0. 9
1, 1, -2, 9, -56, 420, -3572, 33328, -334354, 3559310, -39838760, 465743720, -5658983108, 71191948512, -924554859776, 12365546196641, -169995491295312, 2398380272232272, -34680290150700800, 513390937937217088, -7773229533145403728 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..21.

FORMULA

G.f. satisfies: A(x) = x + x^2*exp( 2*Sum_{n>=0} [d^n/dx^n (x-A(x))^(n+1)/x]/(n+1)! ). - Paul D. Hanna, Mar 24 2011

Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then G(x)^2 = A(x) - x so that G(x + G(x)^2) = x.

EXAMPLE

G.f.: A(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-..;

A(A(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...

The g.f. satisfies:

A(x) = x + x^2*exp((x-A(x))/x + [d/dx (x-A(x))^2/x]/2! + [d^2/dx^2 (x-A(x))^3/x]/3! + [d^3/dx^3 (x-A(x))^4/x]/4! +...)^2.

Higher order iterations of A=A(x) may be expressed in terms of A and x:

A(A(x)) = A + x^2 ;

A(A(A(x))) = (A + A^2) + x^2 ;

A(A(A(A(x)))) = (A + 2*A^2) + (1 + 2*A)*x^2 + x^4 ;

A(A(A(A(A(x))))) = (A + 3*A^2 + 2*A^3 + A^4) + (1 + 4*A + 2*A^2)*x^2 + 2*x^4 ;

A(A(A(A(A(A(x)))))) = (A + 4*A^2 + 6*A^3 + 5*A^4) + (1 + 6*A + 10*A^2 + 8*A^3)*x^2 + (3 + 6*A + 8*A^2)*x^4 + (2 + 4*A)*x^6 + x^8 ;

A(A(A(A(A(A(A(x))))))) = (A + 5*A^2 + 12*A^3 + 18*A^4 + 14*A^5 + 10*A^6 + 4*A^7 + A^8) + (1 + 8*A + 24*A^2 + 40*A^3 + 30*A^4 + 16*A^5 + 4*A^6)*x^2 + (4 + 18*A + 40*A^2 + 24*A^3 + 8*A^4)*x^4 + (6 + 20*A + 8*A^2)*x^6 + 5*x^8 .

The sums of coefficients in the above expansions form A000278: [1,1,2,3,7,16,65,321,4546,107587,20773703,...].

Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then

G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...

G(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...

so that G(x)^2 = A(x) - x and G(x + G(x)^2) = x.

PROG

(PARI) {a(n)=local(A=x+x^2); if(n<1, 0, for(i=3, n, A=A-polcoeff(subst(A, x, A+x*O(x^i))-x^2, i)*x^i); polcoeff(A, n))}

(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(serreverse(A), n))}

(PARI) /* n-th Derivative: */

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

/* G.f.: [Paul D. Hanna, Mar 24 2011] */

{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+x^2*exp(2*sum(m=0, n, Dx(m, (x-A)^(m+1)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}

CROSSREFS

Cf. A000278, A138739.

Cf. A139702, A276370.

Sequence in context: A198953 A212392 A186262 * A276370 A292809 A158883

Adjacent sequences:  A138737 A138738 A138739 * A138741 A138742 A138743

KEYWORD

sign

AUTHOR

Paul D. Hanna, Mar 26 2008, Mar 27 2008, Apr 30 2008

EXTENSIONS

Edited by Paul D. Hanna, May 16 2010

STATUS

approved

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Last modified May 21 18:43 EDT 2019. Contains 323444 sequences. (Running on oeis4.)