A143138 E.g.f.: A(x) = x + (exp(A(x)) - 1)^2.

1, 2, 18, 254, 5010, 126902, 3926538, 143539454, 6053432130, 289293272102, 15450565342938, 911991586990574, 58955877533817810, 4142488437549926102, 314346159031755778218, 25620077133245941688414

Offset

1,2

Comments

Radius of convergence is r = log((2+sqrt(3))/2)/2 - (2-sqrt(3))/2 = 0.17793076... where A(r) = log((sqrt(3)+1)/2) = 0.311905358...

Links

Table of n, a(n) for n=1..16.

Formula

E.g.f. A(x) satisfies:

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

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

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

(4) A'(x) = 1/(1 + 2*exp(A(x)) - 2*exp(2*A(x)) ).

(5) A( log(1+x) - x^2 ) = log(1+x).

a(n) = (n-1)!*(sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(j)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*(-1)^(l-j)*Stirling2(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 08 2012

a(n) ~ sqrt((1-1/sqrt(3))/2) * n^(n-1) / (exp(n) * (sqrt(3)/2 + log((1+sqrt(3))/2) - 1)^(n-1/2)). - Vaclav Kotesovec, Dec 28 2013

Example

A(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5!  + 126902*x^6/6! + 3926538*x^7/7! + 143539454*x^8/8! + 6053432130*x^9/9! + 289293272102*x^10/10! + ...
exp(A(x)) - 1 = G(x) = the g.f. of A143139:
G(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...
Related expansions:
A(x) = x + (exp(x)-1)^2 + d/dx (exp(x)-1)^4/2! + d^2/dx^2 (exp(x)-1)^6/3! + d^3/dx^3 (exp(x)-1)^8/4! + ...
log(A(x)/x) = (exp(x)-1)^2/x + d/dx ((exp(x)-1)^4/x)/2! + d^2/dx^2 ((exp(x)-1)^6/x)/3! + d^3/dx^3 ((exp(x)-1)^8/x)/4! + ...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x-(E^x-1)^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2013 *)

Prog

(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=x + (exp(A)-1)^2); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(serreverse(x-(exp(x+x*O(x^n))-1)^2), n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^(2*m)/m!)); n!*polcoeff(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)); A=x*exp(sum(m=1, n, Dx(m-1, (exp(x+x*O(x^n))-1)^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(Maxima) a(n):=(n-1)!*(sum(binomial(n+k-1, n-1)*sum((-1)^(j)*binomial(k, j)*sum((binomial(j, l)*(2*(j-l))!*(-1)^(l-j)*stirling2(n-l+j-1, 2*(j-l)))/(n-l+j-1)!, l, 0, j), j, 0, k), k, 0, n-1)); /* Vladimir Kruchinin, Feb 08 2012 */

Crossrefs

Cf. A143139.

Sequence in context: A276364 A109517 A213643 * A151362 A215362 A360974

Adjacent sequences: A143135 A143136 A143137 * A143139 A143140 A143141

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2008

Status

approved