Search: id:A000464
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%I A000464 M4812 N2059
%S A000464 1,11,361,24611,2873041,512343611,129570724921,44110959165011,
%T A000464 19450718635716001,10784052561125704811,7342627959965776406281,
%U A000464 6023130568334172003579011,5858598896811701995459355761
%N A000464 Expansion of sin x /cos 2x.
%D A000464 H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate 
               Texts in Mathematics. Springer-Verlag. [From Peter Bala (pbala(AT)talktalk.net), 
               Jun 18 2009]
%D A000464 D. Dumont, Further triangles of Seidel-Arnold type and continued fractions 
               related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 
               275-296.
%D A000464 J. W. L. Glaisher, "On the coefficients in the expansions of cos x/ cos 
               2x and sin x/ cos 2x", Quart. J. Pure and Applied Math., 45 (1914), 
               187-222.
%D A000464 I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.
%D A000464 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 
               689-694; 22 (1968), 699.
%D A000464 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000464 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000464 T. D. Noe, <a href="b000464.txt">Table of n, a(n) for n=0..50</a>
%H A000464 G. E. Andrews, J. Jimenez-Urroz, K. Ono, <a href="http://www.math.wisc.edu/
               ~ono/reprints/055.pdf">q-series identities and values of certain 
               L-functions</a>, Duke Math Jour., Volume 108, No.3 (2001), 395-419. 
               [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]
%F A000464 E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).
%F A000464 a(n)=(-1)^n*L(X,-2n+1) where L(X,z) is the Dirichlet L-function L(X,z)=sum(k=1,
               infty,X(k)/k^z) and where X(k) is the Dirichlet character Legendre(k,
               2) which begins 1,0,-1,0,-1,0,1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0... [From 
               Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 22 2009]
%F A000464 Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 24 2009: (Start)
%F A000464 Basic hypergeometric generating function: 2*exp(-t)*Sum {n = 0..inf} 
               Product {k = 1..n} (1-exp(-16*k*t))/Product {k = 1..n+1} (1+exp(-(16*k-8)*t)) 
               = 1 + 11*t + 361*t^2/2! + 24611*t^3/3! + .... For other sequences 
               with generating functions of a similar type see A000364, A002105, 
               A002439, A079144 and A158690.
%F A000464 a(n) = (-1)^(n+1)*L(-2*n-1), where L(s) is the Dirichlet L-function L(s) 
               = 1 - 1/3^s - 1/5^s + 1/7^s + - - + ... [Andrews et al., Theorem 
               5]. (End)
%F A000464 Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 18 2009: (Start)
%F A000464 a(n) = (-1)^n*B_(2*n+2)(X)/(2*n+2), where B_n(X) denotes the X-Bernoulli 
               number with X a Dirichlet character modulus 8 given by X(8*n+1) = 
               X(8*n+7) = 1, X(8*n+3) = X(8*n+5) = -1 and X(2*n) = 0. See A161722 
               for the values of B_n(X)
%F A000464 For the theory and properties of the generalised Bernoulli numbers B_n(X) 
               and the associated generalised Bernoulli polynomials B_n(X,x) see 
               [Cohen, Section 9.4].
%F A000464 The present sequence also occurs in the evaluation of the finite sum 
               of powers sum {i = 0..m-1} {(8*i+1)^n - (8*i+3)^n - (8*i+5)^n + (8*i+7)^n}, 
               n = 1,2,... - see A151751 for details. (End)
%o A000464 (PARI) a(n)=if(n<0, 0, n+=n+1; n!*polcoeff(sin(x+x*O(x^n))/cos(2*x+x*O(x^n)),
               n)) /* Michael Somos Feb 09 2006 */
%Y A000464 Cf. A064073. Bisection of A000822, A001586.
%Y A000464 Cf. A000364, A002105, A002439, A079144, A158690. [From Peter Bala (pbala(AT)talktalk.net), 
               Mar 24 2009]
%Y A000464 Sequence in context: A067428 A162019 A066268 this_sequence A024149 A018893 
               A051862
%Y A000464 Adjacent sequences: A000461 A000462 A000463 this_sequence A000465 A000466 
               A000467
%K A000464 nonn,easy,nice
%O A000464 0,2
%A A000464 N. J. A. Sloane (njas(AT)research.att.com).
%E A000464 Better description, new reference, Aug 15 1995

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