|
%I #20 Feb 09 2018 03:10:03
%S 1,2,18,254,5010,126902,3926538,143539454,6053432130,289293272102,
%T 15450565342938,911991586990574,58955877533817810,4142488437549926102,
%U 314346159031755778218,25620077133245941688414
%N E.g.f.: A(x) = x + (exp(A(x)) - 1)^2.
%C 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...
%F E.g.f. A(x) satisfies:
%F (1) A(x) = Series_Reversion( x - (exp(x) - 1)^2 ).
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n)/n!.
%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) ((exp(x)-1)^(2*n)/x)/n! ).
%F (4) A'(x) = 1/(1 + 2*exp(A(x)) - 2*exp(2*A(x)) ).
%F (5) A( log(1+x) - x^2 ) = log(1+x).
%F 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
%F 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
%e 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! + ...
%e exp(A(x)) - 1 = G(x) = the g.f. of A143139:
%e G(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ...
%e G(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...
%e Related expansions:
%e 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! + ...
%e 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! + ...
%t Rest[CoefficientList[InverseSeries[Series[x-(E^x-1)^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Dec 28 2013 *)
%o (PARI) {a(n)=local(A=x+O(x^n));for(i=0,n,A=x + (exp(A)-1)^2);n!*polcoeff(A,n)}
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-(exp(x+x*O(x^n))-1)^2),n)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {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)}
%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)); 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)}
%o for(n=1, 25, print1(a(n), ", "))
%o (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 */
%Y Cf. A143139.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 27 2008
| 