A138014 - OEIS

%I #18 Feb 20 2018 02:44:01

%S 1,1,2,16,174,1988,27124,453136,8791980,191869392,4668291000,

%T 125662750464,3706032771336,118759029538368,4109063510399088,

%U 152696171895135744,6065376023980289424,256455323932682550528,11499944141042532006432,545124523779848580648960

%N E.g.f. A(x) satisfies exp(A(x)) = x + exp(A(x)^2) where A(0) = 0.

%H Vincenzo Librandi, <a href="/A138014/b138014.txt">Table of n, a(n) for n = 1..100</a>

%F E.g.f.: A(x) = Series_Reversion( exp(x) - exp(x^2) ).

%F a(n) = -sum(k=1..n-1, (k^n/n!+(k^(n/2)*(-1)^k*((-1)^n+1))/(2*(n/2)!)+sum(j=1..k-1, (-1)^(k-j)*binomial(k,j)*sum(m=1..n, (j^(2*m-n)*(k-j)^(n-m)*binomial(m,n-m))/m!)))*a(k)), n>1, a(1)=1. - _Vladimir Kruchinin_, Jun 25 2011

%F a(n) ~ n^(n-1) * sqrt((2*s-1)/(2-2*s+4*s^2)) / (exp(1+s)*(1-1/(2*s)))^n, where s = 0.6310764773894916166238... is the root of the equation exp(s) = 2*s*exp(s^2). - _Vaclav Kotesovec_, Jan 06 2014

%t Rest[CoefficientList[InverseSeries[Series[E^x-E^(x^2), {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 06 2014 *)

%o (PARI) {a(n)=if(n<1,0,n!*polcoeff(serreverse(exp(x+x*O(x^n))-exp(x^2+x*O(x^n))),n))}

%o (Maxima)

%o   a(n):=if n=1 then 1 else -sum((k^n/n!+(k^(n/2)*(-1)^k*((-1)^n+1))/(2*(n/2)!)+sum((-1)^(k-j)*binomial(k,j)*sum((j^(2*m-n)*(k-j)^(n-m)*binomial(m,n-m))/m!,m,1,n),j,1,k-1))*a(k),k,1,n-1); /* _Vladimir Kruchinin_, Jun 25 2011 */

%K nonn

%O 1,3

%A _Paul D. Hanna_, Feb 27 2008

%E a(19)-a(20) from _Vincenzo Librandi_, Feb 20 2018