cp's OEIS Frontend

A046977 Denominators of Taylor series for sec(x). Also denominators of Taylor series for sech(x) = 1/cosh(x).

%I A046977 #24 Feb 16 2025 08:32:39
%S A046977 1,2,24,720,8064,3628800,95800320,87178291200,4184557977600,
%T A046977 6402373705728000,97316080327065600,1124000727777607680000,
%U A046977 9545360026665222144000,403291461126605635584000000,3209350995912777478963200000,265252859812191058636308480000000
%N A046977 Denominators of Taylor series for sec(x). Also denominators of Taylor series for sech(x) = 1/cosh(x).
%D A046977 G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
%H A046977 Seiichi Manyama, <a href="/A046977/b046977.txt">Table of n, a(n) for n = 0..225</a> (terms 0..100 from T. D. Noe)
%H A046977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Secant.html">Secant</a>
%H A046977 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HyperbolicSecant.html">Hyperbolic Secant</a>
%F A046977 A046976(n)/a(n)= A000364(n)/(2n)!.
%F A046977 Let ZBS(z) = (HurwitzZeta(z,1/4) - HurwitzZeta(z,3/4))/(2^z-2) and R(z) = (cos(z*Pi/2)+sin(z*Pi/2))*(2^z-4^z)*ZBS(1-z)/(z-1)!. Then a(n) = denominator(R(2*n+1)) and A046976(n) = numerator(R(2*n+1)). - _Peter Luschny_, Aug 25 2015
%e A046977 sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + 277/8064*x^8 + 50521/3628800*x^10 + ...
%e A046977 sech(x) = 1 - 1/2 *x^2 + 5/24 *x^4 - 61/720 *x^6 + 277/8064 *x^8 - ...
%p A046977 ZBS := z -> (Zeta(0,z,1/4) - Zeta(0,z,3/4))/(2^z-2):
%p A046977 R := n -> (-1)^floor(n/2)*(2^n-4^n)*ZBS(1-n)/(n-1)!:
%p A046977 seq(denom(R(2*n+1)), n=0..16); # _Peter Luschny_, Aug 25 2015
%t A046977 Table[ EulerE[n]/n! // Denominator, {n, 0, 30, 2}] (* _Jean-François Alcover_, Oct 04 2012 *)
%Y A046977 Cf. A000364, A046976, A099612.
%K A046977 nonn,frac,nice,easy
%O A046977 0,2
%A A046977 _N. J. A. Sloane_