A296436 Expansion of e.g.f. log(1 + arcsin(x))*exp(x).
0, 1, 1, 3, 0, 28, -85, 1029, -6440, 79136, -724305, 9982005, -118974856, 1858582100, -27126378357, 478338929509, -8227405849840, 162502213354272, -3209170996757057, 70409595412300877, -1566861832498793248, 37885426233247176772, -936732798302547171509, 24780850678372964078189
Offset: 0
Keywords
Examples
E.g.f.: A(x) = x/1! + x^2/2! + 3*x^3/3! + 28*x^5/5! - 85*x^6/6! + 1029*x^7/7! - 6440*x^8/8! + ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
Programs
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Maple
a:=series(log(1+arcsin(x))*exp(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019
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Mathematica
nmax = 23; CoefficientList[Series[Log[1 + ArcSin[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]! nmax = 23; CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]! -
PARI
my(ox=O(x^30)); Vecrev(Pol(serlaplace(log(1 + asin(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017
Formula
E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2)))*exp(x), where i is the imaginary unit.
a(n) ~ -(-1)^n * sqrt(Pi) * 2^((n + 1)/2) * n^(n - 1/2) / (exp(n + sin(1)) * (1 - cos(2))^(n/2)). - Vaclav Kotesovec, Dec 21 2017