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

Author

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

Keywords

• nonn

Comments

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

Examples

			G.f. = x^2+1/6*x^4+2/45*x^6+17/1260*x^8+62/14175*x^10+691/467775*x^12+...
		

Links

• 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

Crossrefs

Cf. A000182.

Programs

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 *)

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

Extensions

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