A008775 - OEIS

%I #20 Sep 08 2022 08:44:36

%S 1,2,272,261392,923578112,8687146706432,179207715900772352,

%T 7123449535546491471872,497301503279765920504020992,

%U 56869795869126246818618490355712,10089974849557868979545831504092332032,2659150134955694814127423122143061660925952

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

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

%H G. C. Greubel, <a href="/A008775/b008775.txt">Table of n, a(n) for n = 0..125</a>

%F 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

%p 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

%t 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 *)

%o (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

%o (Magma) m:=70; R<x>:=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

%o (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

%Y Cf. A076417, A101921.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Vaclav Kotesovec_, Sep 15 2014