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A141371 G.f. satisfies: A(x) = x + A(A(A(x))^2). 4
1, 1, 4, 25, 190, 1648, 15700, 160834, 1747372, 19945264, 237585064, 2938737760, 37602695500, 496263658816, 6739460289568, 94002095361937, 1344557294558722, 19696746902333368, 295199862739677892, 4522172757314573464 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Table of n, a(n) for n=1..20.

FORMULA

G.f. A(x) satisfies:

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

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

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

EXAMPLE

G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1648*x^6 + 15700*x^7 +...

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

A(A(x)^2) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 472*x^6 + 4052*x^7 + 38227*x^8 +...

The g.f. satisfies the series:

A(x) = x + A(A(x)^2) + d/dx A(A(x)^2)^2/2! + d^2/dx^2 A(A(x)^2)^3/3! + d^3/dx^3 A(A(x)^2)^4/4! +...

as well as the logarithmic series:

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

PROG

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

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

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

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

{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^2+x*O(x^n))^m)/m!)+x*O(x^n)); polcoeff(A, n)}

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

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

{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^2+x*O(x^n))^m/x)/m!)+x*O(x^n))); polcoeff(A, n)}

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

CROSSREFS

Cf. A141370, A141372.

Sequence in context: A239996 A064063 A171991 * A224079 A199494 A171791

Adjacent sequences:  A141368 A141369 A141370 * A141372 A141373 A141374

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 28 2008

STATUS

approved

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Last modified February 24 22:57 EST 2020. Contains 332216 sequences. (Running on oeis4.)