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

0,2

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.

CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Index entries for Bernoulli numbers B(2n)

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

Example

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+...).

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 *)

Prog

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

Crossrefs

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

Sequence in context: A276294 A066258 A123771 * A188982 A061990 A006742

Adjacent sequences: A046988 A046989 A046990 * A046992 A046993 A046994

Keyword

nonn,easy,frac,nice

Author

N. J. A. Sloane

Status

approved