A296675 Expansion of e.g.f. 1/(1 - arcsinh(x)).

1, 1, 2, 5, 16, 69, 368, 2169, 14208, 109929, 970752, 8995821, 88341504, 988161069, 12276025344, 154843019169, 2009594658816, 29484826539345, 476778061430784, 7588488203093205, 121001549512310784, 2205431202369899925, 44538441694414110720, 852615914764223422665

Offset: 0

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

• sign

Comments

a(48) is negative. - Vaclav Kotesovec, Jan 26 2020

Examples

			1/(1 - arcsinh(x)) = 1 + x/1! + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 69*x^5/5! + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000111, A001818, A006154, A189780, A385343.

Programs

Maple

a:=series(1/(1-arcsinh(x)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 23; CoefficientList[Series[1/(1 - ArcSinh[x]), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[1/(1 - Log[x + Sqrt[1 + x^2]]), {x, 0, nmax}], x] Range[0, nmax]!

PARI

x='x+O('x^99); Vec(serlaplace(1/(1-log(x+sqrt(1+x^2))))) \\ Altug Alkan, Dec 18 2017

Formula

E.g.f.: 1/(1 - log(x + sqrt(1 + x^2))).

a(n) ~ 8*((4 - Pi^2)*sin(Pi*n/2) - 4*Pi*cos(Pi*n/2)) * n^(n-1) / ((4 + Pi^2)^2 * exp(n)). - Vaclav Kotesovec, Dec 18 2017

a(n) = Sum_{k=0..n} k! * i^(n-k) * A385343(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 27 2025