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A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)). 10
0, 1, 1, 1, 17, 31, 691, 10922, 929569, 3202291, 221930581, 9444233042, 56963745931, 29435334228302, 2093660879252671, 344502690252804724, 129848163681107301953, 868320396104950823611, 209390615747646519456961, 28259319101491102261334882 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

REFERENCES

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

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

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-Fran├žois Alcover, Feb 11 2014, after Charles R Greathouse IV *)

PROG

(Sage)

def A046990(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)).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 A058899 A038354

Adjacent sequences:  A046987 A046988 A046989 * A046991 A046992 A046993

KEYWORD

nonn,easy,frac,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 20 08:21 EDT 2019. Contains 326143 sequences. (Running on oeis4.)