OFFSET
1,2
LINKS
Vincenzo Librandi, Jean-François Alcover and Bruno Berselli, Table of n, a(n) for n = 1..100 (up to n = 21 from Vincenzo Librandi, up to n = 40 from Jean-François Alcover)
FORMULA
E.g.f. E(x) satisfies E'/E = y^2, where y=1+x+5*x^2/2+... is defined by y(2-exp(xy))=1.
E.g.f.: exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)).
a(n) ~ c * n^(n-2) / (r^n * exp(n)), where r = 2*(LambertW(2*exp(1))-1)^2 / LambertW(2*exp(1)) = 0.204378273928311464700648197201... and c = 1/((1 - 1/LambertW(2*exp(1))) * exp(1/2)*sqrt(2*(1 + 1/LambertW(2*exp(1))))) = 1.196923669815370203369255598062684... . - Vaclav Kotesovec, Mar 22 2016
MAPLE
seq(n! * coeff(series(exp(int(RootOf(2*_Z-_Z*exp(x*_Z)-1)^2, x)), x, n+1), x, n), n = 1..20); # Vaclav Kotesovec, Mar 21 2016
MATHEMATICA
A000763[max_] := ( e[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 1; c[1] = 1; y[x_] := Sum[d[k]*x^k, {k, 0, max}]; d[0] = 1; d[1] = 1; cc = CoefficientList[ Series[ e'[x]/e[x] - y[x]^2, {x, 0, max}], x]; dd = CoefficientList[ Series[ y[x]*(2 - Exp[x*y[x]]) - 1, {x, 0, max}], x]; eqdd = Thread[dd == 0]; soldd = Solve[ Thread[dd == 0] ]; eqcc = Thread[(cc /. soldd[[1]]) == 0]; solcc = Solve[ Most[eqcc] ] ; solcc /. Rule -> Set; soldd /. Rule -> Set; Table[c[k], {k, 1, max}] *Range[max]! ); Do[A000763[max], {max, 5, 40, 5}]; A000763[40] (* Jean-François Alcover, Jul 23 2013 *)
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Nov 04 2001
STATUS
approved