A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

0, 1, 1, 1, 17, 31, 691, 10922, 929569, 3202291, 221930581, 9444233042, 56963745931, 29435334228302, 2093660879252671, 344502690252804724, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 28259319101491102261334882

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

Crossrefs

Cf. A046991, A002430, A050970.

Programs

Maple

q:= proc(n) add((-1)^k*combinat[eulerian1](n-1,k), k=0..n-1) end: A046990:= n -> numer((-1)^(n-1)*q(2*n)/(2*n)!):
seq(A046990(n),n=0..19);  # Peter Luschny, Nov 16 2012

Mathematica

Join[{0},Numerator[Select[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}], x],#!=0&]]] (* Harvey P. Dale, Jul 27 2011 *)
a[n_] := Numerator[((-4)^n-(-16)^n)*BernoulliB[2*n]/2/n/(2*n)!]; a[0] = 0; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 11 2014, after Charles R Greathouse IV *)

PARI

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

PARI

{a(n) = if( n<1, 0, my(m = 2*n); numerator( polcoeff( -log(cos(x + x * O(x^m))), m)))}; /* Michael Somos, Jun 03 2019 */

Sage

# uses[eulerian1 from A173018]
def A046990(n):
    def q(n):
        return add((-1)^k*eulerian1(n-1, k) for k in (0..n-1))
    return ((-1)^(n-1)*q(2*n)/factorial(2*n)).numer()
[A046990(n) for n in (0..19)]  # Peter Luschny, Nov 16 2012

Formula

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