A046991 - OEIS

%I #27 May 20 2021 09:43:30

%S 1,2,12,45,2520,14175,935550,42567525,10216206000,97692469875,

%T 18561569276250,2143861251406875,34806217964017500,

%U 48076088562799171875,9086380738369043484375,3952575621190533915703125,3920955016221009644377500000,68739242628124575327993046875

%N Denominators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

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

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

%H T. D. Noe, <a href="/A046991/b046991.txt">Table of n, a(n) for n = 0..100</a>

%H Index entries for <a href="/index/Be#Bernoulli">Bernoulli numbers</a> B(2n)

%F A046990(n)/a(n) = 2^(2n-1) *(2^(2n) -1) *abs(B(2n)) / ((2n)! *n).

%F 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

%e log(1/cos(x)) = 1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...

%e log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).

%p 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)!):

%p seq(A046991(n),n=0..17);  # _Peter Luschny_, Nov 16 2012

%t 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_ *)

%t Take[Denominator[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}],x]],{1,-1,2}] (* _Harvey P. Dale_, Jan 18 2020 *)

%o (Sage)

%o def A046991(n):

%o     def q(n):

%o         return add((-1)^k*A173018(n-1, k) for k in (0..n-1))

%o     return ((-1)^(n-1)*q(2*n)/factorial(2*n)).denom()

%o [A046991(n) for n in (0..17)]  # _Peter Luschny_, Nov 16 2012

%o (PARI) a(n)=denominator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ _Charles R Greathouse IV_, Nov 06 2013

%Y Cf. A046990, B(2n) = A027641(2n) / A027642(2n).

%K nonn,easy,frac,nice

%O 0,2

%A _N. J. A. Sloane_