0,2

a(n) is (2n)! times the coefficient of x^(2n) in the Taylor series for cos(x)/cos(2x).

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).

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.

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

cos x / cos 2*x = 1 + 3*x^2/2 + 19*x^4/8 + 307*x^6/80 + ...

(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 */

Cf. A000364 A086646.

Cf. A064069. Bisection of A000825, A001586.

Bisection of A001586. Cf. A098432.

Sequence in context: A012090 A012064 A012204 this_sequence A086220 A127263 A134723

Adjacent sequences: A000278 A000279 A000280 this_sequence A000282 A000283 A000284

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).