A101922 -id:A101922 - cp's OEIS frontend

A263246 Expansion of e.g.f.: sin(rx) / sqrt(1 + cos(rx)^2) where r = sqrt(2), odd powers only.

1, 1, -11, -491, -11159, 460681, 103577629, 8160790429, -624333860399, -386787409545839, -68810049201689531, 6999828208693648549, 9872674440874152431161, 3255253386897615662908441, -346248578699462435167833491, -1072454627614122049417452882131, -584579592415141205182370782224479, 47874474639430619859527348515679521

Offset: 1

Paul D. Hanna, Oct 13 2015

sign

			E.g.f.: S(x) = x + x^3/3! - 11*x^5/5! - 491*x^7/7! - 11159*x^9/9! + 460681*x^11/11! + 103577629*x^13/13! + 8160790429*x^15/15! +...
Related expansions.
S(x)^2 = 2*x^2/2! + 8*x^4/4! - 112*x^6/6! - 9088*x^8/8! - 310528*x^10/10! + 14701568*x^12/12! +...+ -A263249(n)*x^(2*n)/(2*n)! +...
sqrt(1 - S(x)^2) = 1 - x^2/2! - 7*x^4/4! - 49*x^6/6! + 1457*x^8/8! + 148799*x^10/10! + 6409193*x^12/12! +...+ A263247(n)*x^(2*n)/(2*n)! +...
sqrt(1 + S(x)^2) = 1 + x^2/2! + x^4/4! - 71*x^6/6! - 2591*x^8/8! - 23759*x^10/10! + 7872481*x^12/12! +...+ A263248(2*n)*x^(2*n)/(2*n)! +...

G. C. Greubel, Table of n, a(n) for n = 1..234

Cf. A263247, A263248, A263249.

Cf. A101922.

r:= Sqrt[2]; With[{nmax = 500}, CoefficientList[Series[Sin[r*x]/Sqrt[1 + Cos[r*x]^2], {x, 0, nmax}], x]*Range[0, nmax - 1]!][[2 ;; -1 ;; 2]] (* G. C. Greubel, Jul 27 2018 *)

{a(n) = local(S=x,C=1,D=1,ox=O(x^(2*n+2))); for(i=1,2*n+1, S = intformal(C*D^2 +ox); C = 1 - intformal(S*D^2); D = 1 + intformal(S*C*D);); (2*n+1)!*polcoeff(S, 2*n+1)}
for(n=0,20,print1(a(n),", "))

a(n) = - A101922(n). - Michel Marcus, Sep 11 2022

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

Original entry on oeis.org

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

Offset: 0

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

sign

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! -...

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).

Cf. A101922.

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

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

A101924 Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.

Original entry on oeis.org

1, 7, 109, 3163, 166201, 14952367, 2002052389, 353291166403, 77829008955121, 21170264082173527, 7106489649576530269, 2913186117837522604843, 1426879448953133350342441, 816516326741659045770111487, 537701607855913139967684905749, 404270165862091267387117902574483

Offset: 1

Ralf Stephan, Dec 27 2004

Odd coefficients are zero, denominators are 2^n.

Cf. A101922, A012495.

Table[Numerator[(2n)!SeriesCoefficient[ArcCsch[Cos[x]]/Sqrt[2], {x,0,2n}]],{n,14}] (* Stefano Spezia, Aug 29 2022 *)

arccsch(x) = log((1+sqrt(x^2+1))/x);
lista(nn) = localprec(4*nn); my(x='x+O('x^(nn+1)), v=Vec((serlaplace(arccsch(cos(x))))/quadgen(8))); apply(round, vector(#v\2-1, k, v[2*k+1]*2^k)); \\ Michel Marcus, Sep 21 2022

arccsch(cos(x)) = log(sqrt(2)+1) + (1/sqrt(2)) * ((1/2)*x^2/2! + (7/4)*x^4/4! + (109/8)*x^6/6! + (3163/16)*x^8/8! + ...).

arcsech(cos(x)) = Pi/2 - log(sqrt(2)+1) - (1/sqrt(2)) * (-(1/2)*x^2/2! + (7/4)*x^4/4! + (109/8)*x^6/6! + (3163/16)*x^8/8! + ...). [warning: this formula appears to be incorrect since arcsech(cos(0)) = 0; - Michel Marcus, Sep 23 2022]

More terms from Michel Marcus, Sep 20 2022

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A263246 Expansion of e.g.f.: sin(rx) / sqrt(1 + cos(rx)^2) where r = sqrt(2), odd powers only.

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

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A101924 Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.

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cp's OEIS Frontend

A263246 Expansion of e.g.f.: sin(r*x) / sqrt(1 + cos(r*x)^2) where r = sqrt(2), odd powers only.

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

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A101924 Numerators of expansion of e.g.f. 2^(-1/2) * arccsch(cos(x)), even powers only.

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A263246 Expansion of e.g.f.: sin(rx) / sqrt(1 + cos(rx)^2) where r = sqrt(2), odd powers only.