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 A258922 E.g.f. satisfies: A(x) = 1/(3 - 2*exp(x*A(x))). 2
 1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS FORMULA E.g.f. A(x) satisfies: (1) A(x) = (1/x) * Series_Reversion( 3*x - 2*x*exp(x) ). (2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!. (3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ). a(n) = A259063(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015 a(n) ~ (c/3)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015 EXAMPLE E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 302*x^3/3! + 7562*x^4/4! + 253542*x^5/5! +... where A(3*x - 2*x*exp(x)) = 1/(3 - 2*exp(x)). MATHEMATICA CoefficientList[1/x*InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *) PROG (PARI) {a(n) = local(A=1); A = (1/x)*serreverse(3*x - 2*x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)} for(n=0, 20, print1(a(n), ", ")) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)} for(n=0, 25, print1(a(n), ", ")) CROSSREFS Cf. A259063, A052894, A258923. Sequence in context: A121564 A224384 A092563 * A192555 A179497 A227325 Adjacent sequences:  A258919 A258920 A258921 * A258923 A258924 A258925 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 18 2015 STATUS approved

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Last modified December 14 00:21 EST 2017. Contains 295976 sequences.