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A012509 E.g.f.: -log(cos(x)*cos(x)) (even powers only). 2
0, 2, 4, 32, 544, 15872, 707584, 44736512, 3807514624, 419730685952, 58177770225664, 9902996106248192, 2030847773013704704, 493842960380415967232, 140503203207887919775744, 46238368375619195682947072, 17427925514250338592341622784, 7458815407441059142195019251712 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Of course this is 2*log(sec(x)), so a(n) = 2*A000182(n).

LINKS

Table of n, a(n) for n=0..17.

N. J. A. Sloane, Rough notes on Genocchi numbers

FORMULA

G.f.: 2/Q(0) where Q(k) = 1 + x*(2*k + 1)*(2*k + 2)/( -1 + x*(2*k + 2)*(2*k + 3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 11 2013

G.f.: (2/G(0) - 1)*sqrt(-x), where G(k)= 2 + 2*sqrt(-x) - 4*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, May 29 2013

G.f.: 2*x*T(0), where T(k) = 1 - (k+1)*(k+2)*x/((k+1)*(k+2)*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 15 2013

a(n) ~ 2^(2*n+2) * (2*n-1)! / Pi^(2*n). - Vaclav Kotesovec, Feb 08 2015

E.g.f. (odd powers): y = 2*tan(x). - Stanislav Sykora, Nov 28 2016

EXAMPLE

-log(cos(x)*cos(x)) = x^2+1/6*x^4+2/45*x^6+17/1260*x^8+62/14175*x^10+691/467775*x^12+...

MATHEMATICA

nn = 20; Table[(CoefficientList[Series[-Log[Cos[x]^2], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 08 2015 *)

CROSSREFS

Cf. A000182.

Sequence in context: A134958 A101460 A118992 * A062740 A122214 A122216

Adjacent sequences:  A012506 A012507 A012508 * A012510 A012511 A012512

KEYWORD

nonn

AUTHOR

Patrick Demichel (patrick.demichel(AT)hp.com)

EXTENSIONS

Corrected by D. S. McNeil and N. J. A. Sloane, Dec 17 2011 (The signs were wrong and the initial term was missing)

STATUS

approved

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Last modified June 22 23:15 EDT 2017. Contains 288633 sequences.