A296726 Expansion of e.g.f. arcsin(x)/(1 - x).
0, 1, 2, 7, 28, 149, 894, 6483, 51864, 477801, 4778010, 53451135, 641413620, 8446433085, 118250063190, 1792012416075, 28672198657200, 491536207523025, 8847651735414450, 169292834944205175, 3385856698884103500, 71531660838216529125, 1573696538440763640750
Offset: 0
Keywords
Examples
arcsin(x)/(1 - x) = x/1! + 2*x^2/2! + 7*x^3/3! + 28*x^4/4! + 149*x^5/5! + ...
Links
- Muhammad Adam Dombrowski and Gregory Dresden, Areas Between Cosines, arXiv:2404.17694 [math.CO], 2024.
Programs
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Maple
a:=series(arcsin(x)/(1 - x),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019
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Mathematica
nmax = 22; CoefficientList[Series[ArcSin[x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]! nmax = 22; CoefficientList[Series[-I Log[I x + Sqrt[1 - x^2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]! -
PARI
first(n) = x='x+O('x^n); Vec(serlaplace(asin(x)/(1 - x)), -n) \\ Iain Fox, Dec 19 2017
Formula
E.g.f.: -i*log(i*x + sqrt(1 - x^2))/(1 - x), where i is the imaginary unit.
a(n) ~ n! * Pi/2. - Vaclav Kotesovec, Dec 20 2017
a(2*n) = 2*n*a(2*n-1). - Greg Dresden, Apr 04 2024
a(2*n+1) = (2*n+1)*(2*n)*a(2*n-1) + ((2*n-1)!!)^2, using the double factorial notation from A001147. - Greg Dresden, Apr 11 2024