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

0,5

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

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

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: 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 *)

Prog

(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
(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 */

Crossrefs

Cf. A046991, A002430, A050970.

Sequence in context: A276592 A002425 A275994 * A059212 A335360 A058899

Adjacent sequences: A046987 A046988 A046989 * A046991 A046992 A046993

Keyword

nonn,easy,frac,nice

Author

N. J. A. Sloane

Status

approved