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
KEYWORD
nonn,easy,frac,nice
AUTHOR
STATUS
approved