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A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).

%I A046990 #39 Jan 02 2025 12:46:47
%S A046990 0,1,1,1,17,31,691,10922,929569,3202291,221930581,9444233042,
%T A046990 56963745931,29435334228302,2093660879252671,344502690252804724,
%U A046990 129848163681107301953,868320396104950823611,209390615747646519456961,28259319101491102261334882
%N A046990 Numerators of Taylor series for log(1/cos(x)). Also from log(cos(x)).
%D A046990 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 88.
%D A046990 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 42.
%D A046990 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 32, equation 32:6:3 at page 301.
%H A046990 T. D. Noe, <a href="/A046990/b046990.txt">Table of n, a(n) for n = 0..100</a>
%F A046990 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
%e A046990 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 A046990 log(cos(x)) = -(1/2*x^2+1/12*x^4+1/45*x^6+17/2520*x^8+31/14175*x^10+...).
%p A046990 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)!):
%p A046990 seq(A046990(n),n=0..19);  # _Peter Luschny_, Nov 16 2012
%t A046990 Join[{0},Numerator[Select[CoefficientList[Series[Log[1/Cos[x]],{x,0,40}], x],#!=0&]]] (* _Harvey P. Dale_, Jul 27 2011 *)
%t A046990 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_ *)
%o A046990 (Sage) # uses[eulerian1 from A173018]
%o A046990 def A046990(n):
%o A046990     def q(n):
%o A046990         return add((-1)^k*eulerian1(n-1, k) for k in (0..n-1))
%o A046990     return ((-1)^(n-1)*q(2*n)/factorial(2*n)).numer()
%o A046990 [A046990(n) for n in (0..19)]  # _Peter Luschny_, Nov 16 2012
%o A046990 (PARI) a(n)=numerator(((-4)^n-(-16)^n)*bernfrac(2*n)/2/n/(2*n)!) \\ _Charles R Greathouse IV_, Nov 06 2013
%o A046990 (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 */
%Y A046990 Cf. A046991, A002430, A050970.
%K A046990 nonn,easy,frac,nice
%O A046990 0,5
%A A046990 _N. J. A. Sloane_