A012495 Expansion of e.g.f. arcsinh(sin(x)) (odd powers only).

1, -2, 20, -632, 39440, -4087712, 634237760, -137605112192, 39776178356480, -14775064298435072, 6857795892626969600, -3889298341511511652352, 2646362625886738240901120, -2127690488032789501903020032

Offset: 0

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Keywords

• sign

Comments

arcsinh(cos(x)*tan(x)) = x - 2/3!*x^3 + 20/5!*x^5 - 632/7!*x^7 + 39440/9!*x^9...

arcsin(sinh(x)) = x + 2*x^3/3! + 20*x^5/5! + 620*x^7/7! +...

arccosh(sin(x)) = Pi/2 - x + 2*x^3/3! - 20*x^5/5! + 620*x^7/7! -...

Links

• Nikolay Sh. Izmailian, Ralph Kenna, and Vladimir V. Papoyan, Exact coefficients of finite-size corrections in the Ising model with Brascamp-Kunz boundary conditions and their relationships for strip and cylindrical geometries, arXiv:2303.03484 [cond-mat.stat-mech], 2023. See also J. Phys. A: Math. Theor, (2023).

Crossrefs

Cf. A101922.

Programs

Mathematica

Table[n!*SeriesCoefficient[ArcSinh[Sin[x]],{x,0,n}],{n,1,40,2}] (* Vaclav Kotesovec, Oct 30 2013 *)

Formula

a(n) ~ (-1)^n * 2^(2*n+5/4)*n^(2*n) / (exp(2*n)*log(1+sqrt(2))^(2*n+1/2)). - Vaclav Kotesovec, Oct 30 2013