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A046991 Denominators of Taylor series for log(1/cos(x)). Also from log(cos(x)). 3
1, 2, 12, 45, 2520, 14175, 935550, 42567525, 10216206000, 97692469875, 18561569276250, 2143861251406875, 34806217964017500, 48076088562799171875, 9086380738369043484375, 3952575621190533915703125, 3920955016221009644377500000, 68739242628124575327993046875 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
LINKS
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
KEYWORD
nonn,easy,frac,nice
AUTHOR
STATUS
approved

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Last modified July 18 06:54 EDT 2024. Contains 374377 sequences. (Running on oeis4.)