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 A259062 E.g.f.: Series_Reversion( -x + 2*x*exp(-x) ). 2
 1, 4, 42, 728, 17630, 548532, 20852370, 936655792, 48540537702, 2850727359500, 187107038833946, 13572973331551944, 1078343465147156910, 93119965280416893028, 8684514946963752624930, 869915871265946242868576, 93146889134541855185069942, 10617155946603647157142073916 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Table of n, a(n) for n=1..18. FORMULA E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!. E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ). a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^n * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - Vaclav Kotesovec, Jun 19 2015 EXAMPLE E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 728*x^4/4! + 17630*x^5/5! +... where A(-x + 2*x*exp(-x)) = x. MATHEMATICA Rest[CoefficientList[InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *) PROG (PARI) {a(n) = local(A=x); A = serreverse(-x + 2*x*exp(-x +x*O(x^n) )); n!*polcoeff(A, n)} for(n=1, 20, print1(a(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, 2^m*(1-exp(-x+x*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)} for(n=1, 25, print1(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, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=1, 25, print1(a(n), ", ")) CROSSREFS Cf. A258872, A053492, A259063, A259064, A259065, A259066. Sequence in context: A192949 A156453 A327200 * A074768 A295763 A266526 Adjacent sequences: A259059 A259060 A259061 * A259063 A259064 A259065 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 18 2015 STATUS approved

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Last modified September 8 09:26 EDT 2024. Contains 375753 sequences. (Running on oeis4.)