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A000464
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Expansion of sin x /cos 2x.
(Formerly M4812 N2059)
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11
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1, 11, 361, 24611, 2873041, 512343611, 129570724921, 44110959165011, 19450718635716001, 10784052561125704811, 7342627959965776406281, 6023130568334172003579011, 5858598896811701995459355761
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| 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. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
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.
I. J. Schwatt, Intro. to Operations with Series, Chelsea, p. 278.
D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Vieru, Agoh's conjecture: its proof, its generalizations, its analogues, Arxiv preprint arXiv:1107.2938, 2011.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..50
G. E. Andrews, J. Jimenez-Urroz, K. Ono, q-series identities and values of certain L-functions, Duke Math Jour., Volume 108, No.3 (2001), 395-419. [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]
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FORMULA
| E.g.f.: Sum_{k>=0} a(k)x^(2k+1)/(2k+1)! = sin(x)/cos(2x).
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]
Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 24 2009: (Start)
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.
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)
Contribution from Peter Bala (pbala(AT)talktalk.net), Jun 18 2009: (Start)
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)
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].
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)
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PROG
| (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 */
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CROSSREFS
| Cf. A064073. Bisection of A000822, A001586.
Cf. A000364, A002105, A002439, A079144, A158690. [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]
Sequence in context: A162019 A066268 A176474 * A024149 A018893 A051862
Adjacent sequences: A000461 A000462 A000463 * A000465 A000466 A000467
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Better description, new reference, Aug 15 1995
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