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

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

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

Crossrefs

Cf. A076417, A101921.

Programs

Magma

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

Maple

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

Mathematica

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 *)

PARI

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

Sage

[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

Formula

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

Extensions

More terms from Vaclav Kotesovec, Sep 15 2014