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A000281
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Expansion of cos(x)/cos(2x).
(Formerly M3163 N1281)
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7
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1, 3, 57, 2763, 250737, 36581523, 7828053417, 2309644635483, 898621108880097, 445777636063460643, 274613643571568682777, 205676334188681975553003, 184053312545818735778213457, 193944394596325636374396208563
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n) is (2n)! times the coefficient of x^(2n) in the Taylor series for cos(x)/cos(2x).
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REFERENCES
| 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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..50
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6, 2003.
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FORMULA
| a(n) = Sum_{k=0..n} (-1)^k*binomial(2n, 2k)*A000364(n-k)*4^(n-k). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 26 2004
E.g.f.: Sum_{k>=0} a(k)x^(2k)/(2k)! = cos(x)/cos(2x).
a(n) is approximately 2^(4*n-3)*(2*n-1)!*sqrt(2)/((Pi^(2*n-1))*(2*n-1)). The approximation is quite good a(250) is of the order of 10^1181 and this formula is accurate to 238 digits. - Simon Plouffe, Jan 31 2007
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EXAMPLE
| cos x / cos 2*x = 1 + 3*x^2/2 + 19*x^4/8 + 307*x^6/80 + ...
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MATHEMATICA
| With[{nn=30}, Take[CoefficientList[Series[Cos[x]/Cos[2x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* From Harvey P. Dale, Oct 06 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, n*=2; n!*polcoeff(cos(x+x*O(x^n))/cos(2*x+x*O(x^n)), n)) /* Michael Somos Feb 09 2006 */
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CROSSREFS
| Cf. A000364 A086646.
Cf. A064069. Bisection of A000825, A001586.
Bisection of A001586. Cf. A098432.
Sequence in context: A012090 A012064 A012204 * A086220 A127263 A134723
Adjacent sequences: A000278 A000279 A000280 * A000282 A000283 A000284
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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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