A076417 -id:A076417 - cp's OEIS frontend

A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

1, 1, 6, 71, 1456, 45541, 2020656, 120686411, 9336345856, 908138776681, 108480272749056, 15611712012050351, 2664103110372192256, 531909061958526321421, 122840808510269863827456, 32491881630252866646683891, 9758611490955498257378246656

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..100
Nicolas Basset, Counting and generating permutations using timed languages, 2013.
Nicolas Basset, Counting and generating permutations in regular classes of permutations, HAL Id: hal-01093994, 2014.
B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002; Moscow Math. J., 3 (2003), 647-659.
P. R. Stein & N. J. A. Sloane, Correspondence, 1975
Eric Weisstein's World of Mathematics, Generalized Hyperbolic Functions

Bisection of A058258.

Cf. A131453, A131454, A131455, A076417.

g:=((cosh(x)-1)*sin(x)+(cos(x)+1)*sinh(x))/(cos(x)*cosh(x)+1): series(%,x,35):
seq(n!*coeff(%,x,n),n=1..34,2); # Peter Luschny, Feb 07 2017

egf = ((Cosh[x]-1)*Sin[x]+(Cos[x]+1)*Sinh[x])/(Cos[x]*Cosh[x]+1); a[n_] := SeriesCoefficient[egf, {x, 0, 2*n+1}]*(2*n+1)!; Array[a, 17, 0] (* Jean-François Alcover, Mar 13 2014 *)

E.g.f.: x + Sum_{n>=1} a(n)*(x^(2n+1))/(2n+1)! = (f(0,x)*f(1,x) -f(2,x)*f(3,x)+ f(3,x))/(f(0,x)^2 - f(1,x)*f(3,x)), where f(j,x) = Sum_{k>=0} (x^(4k+j))/(4k+j)!, j = 0,1,2,3, is the j-th generalized hyperbolic function. - Peter Bala, Jul 13 2007
Basset (2013) gives an e.g.f. involving trigonometric and hyperbolic functions. - N. J. A. Sloane, Dec 24 2013
a(n) ~ 4 * (2*n+1)! / (tan(r/2)^2 * r^(2*n+2)), where r = A076417 = 1.8751040687119611664453082410782141625701117335310699882454137131... is the root of the equation cos(r)*cosh(r) = -1. - Vaclav Kotesovec, Feb 01 2015

A058258 The 2-Up sequence: formed from final entries in rows of A058257.

Original entry on oeis.org

1, 1, 1, 1, 3, 6, 26, 71, 413, 1456, 10576, 45541, 397023, 2020656, 20551376, 120686411, 1402815833, 9336345856, 122087570176, 908138776681, 13194844482843, 108480272749056, 1733786041150976, 15611712012050351, 272197308765744053, 2664103110372192256

Offset: 0

N. J. A. Sloane, Dec 06 2000

Alois P. Heinz, Table of n, a(n) for n = 0..200
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.

Cf. A005981, A076417.

Column k=2 of A229892.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(`if`(t=2,
       b(o-j, u+j-1, 1), b(u+j-1, o-j, t+1)), j=1..o))
    end:
a:= n-> b(0, n, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 02 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[If[t == 2, b[o-j, u+j-1, 1], b[u+j-1, o-j, t+1]], {j, 1, o}]] ;a[n_] := b[0, n, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 03 2014, after Alois P. Heinz *)
CoefficientList[Series[1 + ((Sin[x]-Cos[x]+1) * (Cosh[x]-1) + (Sin[x]+Cos[x]+1) * Sinh[x]) / ((1+Cosh[x]*Cos[x])),{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 06 2014 *)

E.g.f. (J. M. Luck, 2013): 1 + ((sin(x) - cos(x) + 1) * (cosh(x)-1) + (sin(x) + cos(x) + 1) * sinh(x)) / ((1 + cosh(x)*cos(x))). - Vaclav Kotesovec, Sep 06 2014

a(n) ~ c * n! / r^n, where r = A076417 = 1.8751040687119611664453... is the root of the equation cosh(r)*cos(r) = -1, and c = 4*cot(r/2)/r = 1.56598351207925... if n is even, c = 4*cot(r/2)^2/r = 1.14958147083780... if n is odd. - Vaclav Kotesovec, Sep 06 2014

More terms from Larry Reeves (larryr(AT)acm.org), Dec 12 2000

A008775 Expansion of e.g.f. 2/(1 + cos x * cosh x) in powers of x^4.

Original entry on oeis.org

1, 2, 272, 261392, 923578112, 8687146706432, 179207715900772352, 7123449535546491471872, 497301503279765920504020992, 56869795869126246818618490355712, 10089974849557868979545831504092332032, 2659150134955694814127423122143061660925952

Offset: 0

N. J. A. Sloane

nonn

Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 287.

G. C. Greubel, Table of n, a(n) for n = 0..125

Cf. A076417, A101921.

m:=70; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 2/(1+Cos(x)*Cosh(x)) )); [Factorial(4*n-4)*b[4*n-3]: n in [1..Floor(m/4)]]; // G. C. Greubel, Sep 11 2019

seq(factorial(4*n)*coeff(series(2/(1+cos(x)*cosh(x)), x, 4*n+1), x, 4*n), n = 0..15); # G. C. Greubel, Sep 11 2019

Table[(CoefficientList[Series[2/(1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 60]! )[[n]], {n, 1, 61, 4}] (* Vaclav Kotesovec, Sep 15 2014 *)

my(x='x+O('x^60)); v=Vec(serlaplace(2/(1+cos(x)*cosh(x)) )); vector(#v\4, n, v[4*n-3]) \\ G. C. Greubel, Sep 11 2019

[factorial(4*n)*( 2/(1+cos(x)*cosh(x)) ).series(x,4*n+2).list()[4*n] for n in (0..30)]; # G. C. Greubel, Sep 11 2019

a(n) ~ 8 * (4*n)! / ((cosh(r)*sin(r) - cos(r)*sinh(r)) * r^(4*n+1)), where r = 1.875104068711961166445308241... is the root of the equation cosh(r)*cos(r) = -1. See A076417. - Vaclav Kotesovec, Sep 15 2014

More terms from Vaclav Kotesovec, Sep 15 2014

A009398 Expansion of log(1+tanh(x)*tan(x)).

Original entry on oeis.org

0, 2, -12, 352, -16352, 1382912, -172372992, 30716649472, -7334631698432, 2270814362796032, -883463096359452672, 422158497266733678592, -243014206258719292915712, 165880670114405086398513152

Offset: 0

R. H. Hardin

sign

Cf. A076417.

Log[ 1+Tanh[ x ]*Tan[ x ] ] (* Even Part *)
nn = 20; Table[(CoefficientList[Series[Log[1 + Tan[x]*Tanh[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Jan 24 2015 *)

a(n) ~ (2*n)! * (-1)^(n+1) * 2^(2*n) / (n * r^(2*n)), where r = 1.87510406871196116644530824107821416257011173353... (see A076417) is the root of the equation sin(r) = tanh(r). - Vaclav Kotesovec, Jan 24 2015

Extended with signs by Olivier Gérard, Mar 15 1997

A076418 Decimal expansion of second solution of equation cos(x)*cosh(x) = -1.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Oct 10 2002

This is an equation related to a cantilever beam: cos(x)*cosh(x) = -1. The first three solutions are: 1.875 (A076417), 4.69409 (this sequence) and 7.854757 (A076419).

			4.694091132974174576436391778...

Z. Guede, I. Elishakov, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos, Solitons and Fractals 12 (7) (2001) 1267-1298.

Cf. A076417, A076419.

RealDigits[x/.FindRoot[Cos[x]Cosh[x]==-1,{x,4},WorkingPrecision-> 120]] [[1]] (* Harvey P. Dale, May 01 2019 *)

solve(x=4, 5, cos(x)*cosh(x) + 1) \\ Michel Marcus, Sep 11 2019

A076419 Decimal expansion of third solution of equation cos(x)*cosh(x) = -1.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Oct 10 2002

This is an equation related to a cantilever beam: cos(x)*cosh(x) = -1. The first three solutions are: 1.875 (A076417), 4.69409 (A076418) and 7.854757 (this sequence).

			cos(x) cosh(x) = -1, x = 7.854757...

Z. Guede, I. Elishakov, A fifth-order polynomial that serves as both buckling and vibration mode of an inhomogeneous structure, Chaos, Solitons and Fractals 12 (7) (2001) 1267-1298.

Cf. A076417, A076418.

RealDigits[x/.FindRoot[Cos[x]Cosh[x]==-1,{x,8},WorkingPrecision-> 120]] [[1]] (* Harvey P. Dale, Oct 17 2021 *)

solve(x=7, 8, cos(x)*cosh(x) + 1) \\ Michel Marcus, Sep 11 2019

A068353 Decimal expansion of smallest positive root of cos(Pix/2)cosh(Pi*x/2) = -1.

Original entry on oeis.org

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

Offset: 1

Joost de Winter, Feb 28 2002

			1.1937283253889343234388...

E. A. Harrington, The vibrations of tuning-forks, J. Opt. Soc. Am., 17, 224-237 (1928).
G. H. Keulegan, On the vibration of U bars, Doctoral dissertation, Johns Hopkins University, 1928.
A. M. Liebetrau, Some tests of randomness based upon the variance-time curve of the Poisson process Journal of the Royal Statistical Society, Series B (Methodological), 41, September 1979, 32-39.

Equals (2/Pi) times A076417.

RealDigits[FindRoot[Cos[Pi x/2]Cosh[Pi x/2]==-1, {x, 1}, WorkingPrecision->200][[1, 2]]][[1]]

solve(x=1,2,cos(x)*cosh(x)+1)*2/Pi \\ Charles R Greathouse IV, Apr 16 2014

Edited by Dean Hickerson, Jun 07 2002

Offset corrected by R. J. Mathar, Feb 05 2009

cp's OEIS Frontend

A005981 Number of 2 up, 2 down, 2 up, ... permutations of length 2n + 1.

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A058258 The 2-Up sequence: formed from final entries in rows of A058257.

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A008775 Expansion of e.g.f. 2/(1 + cos x * cosh x) in powers of x^4.

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Magma

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Sage

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A009398 Expansion of log(1+tanh(x)*tan(x)).

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A076418 Decimal expansion of second solution of equation cos(x)*cosh(x) = -1.

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A076419 Decimal expansion of third solution of equation cos(x)*cosh(x) = -1.

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A068353 Decimal expansion of smallest positive root of cos(Pi*x/2)*cosh(Pi*x/2) = -1.

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Mathematica

PARI

Extensions

A068353 Decimal expansion of smallest positive root of cos(Pix/2)cosh(Pi*x/2) = -1.