A200904 - OEIS

%I #20 Oct 13 2017 05:12:32

%S 1,0,-6,-24,170,3420,7126,-612528,-9539982,84591300,5635854158,

%T 50835366648,-2827700267990,-99260514667860,438301096424070,

%U 118700423262570144,2500018105640527586,-100537942668123244140,-6930391127610546125378,-31482610140761483476920

%N E.g.f. A(x) satisfies: A(x) = x*(2*exp(A(x)) - exp(2*A(x))).

%H Vaclav Kotesovec, <a href="/A200904/b200904.txt">Table of n, a(n) for n = 1..390</a>

%F a(n) = Sum_{j=0..n} 2^j*(2*n-j)^(n-1)*(-1)^(n-j)*binomial(n,j), n > 0, a(0)=0.

%F Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(1/r) = 2.35574803651010217..., where r = 0.14939557933379789490 - 0.39733586552717789127*i is the complex root of the equation sqrt((r-2)*r) + 2*log(1/2 - sqrt((r-2)*r)/(2*r)) = 1+r, and i is the imaginary unit. - _Vaclav Kotesovec_, Jan 26 2014

%t Rest[CoefficientList[InverseSeries[Series[-(x/(E^x*(-2 + E^x))),{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 26 2014 *)

%t Table[Sum[2^j*(2*n-j)^(n-1)*(-1)^(n-j)*Binomial[n,j],{j,0,n}],{n,1,20}] (* _Vaclav Kotesovec_ after _Vladimir Kruchinin_, Jan 26 2014 *)

%o (Maxima)

%o a(n):=(sum(2^j*(2*n-j)^(n-1)*(-1)^(n-j)*binomial(n,j),j,0,n));

%Y Cf. A236466.

%K sign

%O 1,3

%A _Vladimir Kruchinin_, Nov 23 2011