A046991 Denominators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

1, 2, 12, 45, 2520, 14175, 935550, 42567525, 10216206000, 97692469875, 18561569276250, 2143861251406875, 34806217964017500, 48076088562799171875, 9086380738369043484375, 3952575621190533915703125, 3920955016221009644377500000, 68739242628124575327993046875

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• frac
• nice

Examples

			log(1/cos(x)) = 1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...
log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
• CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
• Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:3 at page 301.

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• Index entries for Bernoulli numbers B(2n)

Crossrefs

Cf. A046990, B(2n) = A027641(2n) / A027642(2n).

Programs

Maple

q:= proc(n) add((-1)^k*combinat[eulerian1](n-1,k), k=0..n-1) end: A046991:= n -> denom((-1)^(n-1)*q(2*n)/(2*n)!):
seq(A046991(n),n=0..17);  # Peter Luschny, Nov 16 2012

Mathematica

a[n_] := Denominator[((-4)^n-(-16)^n)*BernoulliB[2*n]/2/n/(2*n)!]; a[0] = 0; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Feb 11 2014, after Charles R Greathouse IV *)
Take[Denominator[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}],x]],{1,-1,2}] (* Harvey P. Dale, Jan 18 2020 *)

PARI

a(n)=denominator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ Charles R Greathouse IV, Nov 06 2013

Sage

def A046991(n):
    def q(n):
        return add((-1)^k*A173018(n-1, k) for k in (0..n-1))
    return ((-1)^(n-1)*q(2*n)/factorial(2*n)).denom()
[A046991(n) for n in (0..17)]  # Peter Luschny, Nov 16 2012

Formula

A046990(n)/a(n) = 2^(2n-1) *(2^(2n) -1) *abs(B(2n)) / ((2n)! *n).

Let q(n) = Sum_{k=0..n-1} (-1)^k*A201637(n-1,k) then a(n) = denominator((-1)^(n-1)*q(2*n)/(2*n)!). - Peter Luschny, Nov 16 2012