A009379 -id:A009379 - cp's OEIS frontend

A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

1, 1, 9, 9, 6, 7, 8, 6, 4, 0, 2, 5, 7, 7, 3, 3, 8, 3, 3, 9, 1, 6, 3, 6, 9, 8, 4, 8, 6, 4, 1, 1, 4, 1, 9, 4, 4, 2, 6, 1, 4, 5, 8, 7, 8, 8, 4, 1, 8, 6, 0, 7, 2, 0, 8, 9, 1, 5, 4, 7, 7, 7, 8, 3, 9, 1, 8, 1, 2, 4, 7, 2, 5, 2, 2, 3, 8, 4, 7, 4, 7, 9, 9, 9, 9, 0, 8, 6, 9, 9, 2, 1, 4, 6, 5, 0, 9, 3, 7, 9, 8, 8

Offset: 1

Eric W. Weisstein, Jul 06 2003

This constant can also be defined as the root of coth x = x, as this equation and the above are equivalent. - Carl R. White, Dec 09 2003. Also the root of x*tanh x = 1. - N. J. A. Sloane, May 07 2020
This constant is also the point on the parametric tractrix (t - tanh(t), sech(t)) the least distant from the origin. - Michael Clausen, Feb 18 2013
This constant also equals sqrt(lambda^2+1), where lambda is the Laplace limit constant A033259. - Jean-François Alcover, Sep 08 2014, after Steven Finch.
For each of the real symmetric n X n matrices M defined by M(i,j) = max(i,j) with n >= 2, there exist n-1 negative eigenvalues < -1/4 and only one positive eigenvalue lambda(n) such that n^2/2 < lambda(n) < n^2. Indeed, when n tends to infinity, lambda(n) ~ n^2/(this constant)^2 (see reference O. Carton et al.). For n = 2, the positive eigenvalue is (3+sqrt(17))/2 [A178255]. - Bernard Schott, Mar 13 2020

			1.1996786402577338339163698486411419442614587884186072...

O. Carton, L. Rosaz, M. Zeitoun, Problèmes corrigés de Mathématiques posés au Concours de Mines/Ponts, Tome 5, Ellipses, 1992; Problème Mines-Ponts 1991 - Options M, P', TA - Epreuve pratique p. 125.
Steven R. Finch, Mathematical constants, Volume 94, Encyclopedia of mathematics and its applications, Cambridge University Press, 2003, p. 268.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 30, equation 30:10:9 at page 283.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Jogundas Armaitis, Molecules and Polarons in Extremely Imbalanced Fermi Mixtures, Master's Thesis, Aug 11 2011, Institute for Theoretical Physics, Utrecht University.
Robert Ferréol, Tractrix, Mathcurve.
D. E. Knuth, Whirlpool Permutations, May 05 2020.
Mathematics Stack Exchange, Laplace limit constant.
Eric Weisstein's World of Mathematics, Kepler's Equation.
Eric Weisstein's World of Mathematics, Laplace Limit.
Eric Weisstein's World of Mathematics, Hyperbolic Cotangent
Wikipedia, Tractrix.
Index entries for transcendental numbers.

Cf. A003957 (x = cos(x)), A009379, A033259, A069855, A209289.

RealDigits[ x /. FindRoot[ Coth[x] == x, {x, 1}, WorkingPrecision -> 102]] // First (* Jean-François Alcover, Feb 08 2013 *)
1+2 NSum[LaguerreL[n-1,1,4 n]/n Exp[-2 n],{n,1,Infinity}] //
  (* Aaron Hendrickson, Mar 17 2021 *)

solve(u=1,2,tanh(u)-1/u)  /* type e.g. \p99 to get 99 digits; M. F. Hasler, Feb 01 2011 */

Equals 1 + 2*Sum_{n>=1} (Laguerre(n-1,1,4n)/n)*e^(-2n) (see Mathematics Stack Exchange in Links). - Aaron Hendrickson, Mar 17 2022

A009399 Expansion of log(1+tanh(x)*x).

Original entry on oeis.org

0, 2, -20, 576, -33312, 3258880, -485139456, 102300807168, -29028932390912, 10668077137133568, -4929291212351078400, 2797060130323340197888, -1912137417504544127975424, 1550018044651811766917922816

Offset: 0

R. H. Hardin

sign

Cf. A009379, A069855, A085984.

With[{nn=30},Take[CoefficientList[Series[Log[1+Tanh[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jun 13 2016 *)

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = A069855 = 0.8603335890193797624838934241376623334118843632... is the root of the equation r * tan(r) = 1. - Vaclav Kotesovec, Dec 21 2017

Extended with signs by Olivier Gérard, Mar 15 1997

Prior Mathematica program replaced by Harvey P. Dale, Jun 13 2016

A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -2, 18, -312, 9470, -436860, 28616322, -2522596496, 288046961190, -41355026494020, 7291524732108650, -1548849359704927896, 390122366308850972238, -114968364853645904762252, 39189956630839558368115410, -15300235972710835734174638880

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tan(x/2)) = x^2/2! - 2*x^4/4! + 18*x^6/6! - 312*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296838.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tan(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.54340463841820844795870974005331555369788376471926269... is the root of the equation r*tanh(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

Original entry on oeis.org

0, 1, -4, 48, -1186, 50060, -3226206, 294835184, -36270477034, 5779302944436, -1157856177719830, 284876691727454552, -84442374415240892898, 29680054107768128647388, -12205478262363331593956686, 5805823539844285054558025280, -3163004294186696659107788567386

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

sign

			log(1 + x*tanh(x/2)) = x^2/2! - 4*x^4/4! + 48*x^6/6! - 1186*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A001469, A003707, A009379, A009399, A110501, A296837.

nmax = 16; Table[(CoefficientList[Series[Log[1 + x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] log(1 + x*tanh(x/2)).

a(n) ~ -(-1)^n * sqrt(Pi) * 2^(2*n + 1) * n^(2*n - 1/2) / (r^(2*n) * exp(2*n)), where r = 1.306542374188806202228727831923118284841279755635... is the root of the equation r * tan(r/2) = 1. - Vaclav Kotesovec, Dec 21 2017

A024236 Expansion of log(1+tan(x)*x)/2.

Original entry on oeis.org

0, 1, -2, 48, -1040, 62720, -4314624, 498003968, -70081316864, 13529482592256, -3175495421788160, 933402531586899968, -326784893253162369024, 136568449424117316190208, -66517446960602151366164480

Offset: 0

R. H. Hardin

sign

Cf. A009379.

Cf. A101921.

S:= series(log(1+tan(x)*x)/2, x, 101): seq(coeff(S, x, 2*j)*(2*j)!, j=0..50); # Robert Israel, Jun 29 2015

With[{nn=30},Take[CoefficientList[Series[Log[1+Tan[x]x]/2,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jun 24 2018 *)

E.g.f. of aerated sequence: Sum(n>=0, a(n)*x^(2*n)/(2*n)!) = log(1+tan(x)*x)/2. - Robert Israel, Jun 29 2015

Extended with signs, Mar 1997

Previous Mathematica program replaced by Harvey P. Dale, Jun 24 2018

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A085984 Decimal expansion of solution to e^x*(-1 + x) = (1 + x)/e^x.

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A009399 Expansion of log(1+tanh(x)*x).

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A296837 Expansion of e.g.f. log(1 + x*tan(x/2)) (even powers only).

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A296838 Expansion of e.g.f. log(1 + x*tanh(x/2)) (even powers only).

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A024236 Expansion of log(1+tan(x)*x)/2.

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