A296437 Expansion of e.g.f. log(1 + arcsinh(x))*exp(x).
0, 1, 1, 1, 0, 8, -5, -51, -504, 8224, -12865, -296155, -2166736, 73348780, -116217309, -7440979651, -39733320080, 2564082122752, -3056854891489, -544155777899859, -2138400746459448, 251904027415707852, -163714875656114029, -92626483427571793931, -273784346863222483272
Offset: 0
Keywords
Examples
E.g.f.: A(x) = x/1! + x^2/2! + x^3/3! + 8*x^5/5! - 5*x^6/6! - 51*x^7/7! - 504*x^8/8! + ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- Vaclav Kotesovec, Graph - The asymptotic ratio
Programs
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Maple
a:=series(log(1+arcsinh(x))*exp(x),x=0,25): seq(n!*coeff(a,x,n),n=0..24); # Paolo P. Lava, Mar 27 2019
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Mathematica
nmax = 24; CoefficientList[Series[Log[1 + ArcSinh[x]] Exp[x], {x, 0, nmax}], x] Range[0, nmax]! nmax = 24; CoefficientList[Series[Log[1 + Log[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 + asinh(x + ox)) * exp(x + ox)))) \\ Andrew Howroyd, Dec 12 2017
Formula
E.g.f.: log(1 + log(x + sqrt(1 + x^2)))*exp(x).
a(n) ~ n! * 2*sqrt(2/Pi) * (Pi*c - 2*s) / (n^(3/2) * (4 + Pi^2)) * (1 + (c*(-192 + 208*Pi - 96*Pi^2 - 8*Pi^3 - 12*Pi^4 + Pi^5) - 2*s*(80 + 48*Pi - 40*Pi^2 + 24*Pi^3 + Pi^4 + 3*Pi^5)) / (4*(4 + Pi^2)^2 * (c*Pi - 2*s)*n)), where s = sin(1 - Pi*n/2) and c = cos(1 - Pi*n/2). - Vaclav Kotesovec, Dec 21 2017