A012509 E.g.f.: -log(cos(x)*cos(x)) (even powers only).

0, 2, 4, 32, 544, 15872, 707584, 44736512, 3807514624, 419730685952, 58177770225664, 9902996106248192, 2030847773013704704, 493842960380415967232, 140503203207887919775744, 46238368375619195682947072, 17427925514250338592341622784, 7458815407441059142195019251712

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.

Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, arXiv:1202.1203 [math.NT], 2012. See p. 21.

Tewodros Amdeberhan, Victor H. Moll and Christophe Vignat, A probabilistic interpretation of a sequence related to Narayana Polynomials, Online Journal of Analytic Combinatorics, Issue 8, 2013. See p. 21.

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

G.f. = 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: A101460 A304862 A118992 * A062740 A336832 A122214

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