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A012509 E.g.f.: -log(cos(x)*cos(x)) (even powers only). 3
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
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.
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
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 July 20 18:59 EDT 2024. Contains 374459 sequences. (Running on oeis4.)