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A000816 E.g.f.: Sum_{n >= 0} a(n) * x^(2*n) / (2*n)! = sin(x)^2 / cos(2*x). 6
0, 2, 40, 1952, 177280, 25866752, 5535262720, 1633165156352, 635421069967360, 315212388819402752, 194181169538675507200, 145435130631317935357952, 130145345400688287667978240, 137139396592145493713802493952 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..200

FORMULA

1/2 * A002436(n), n>0. - Ralf Stephan, Mar 09 2004

a(n) = 2^(2*n - 1) * A000364(n) except n=0.

E.g.f.: sin(x)^2/cos(2x)=1/Q(0)-1/2;   Q(k)=1+1/(1-2*(x^2)/(2*(x^2)-(k+1)*(2k+1)/Q(k+1)));  (continued fraction). - Sergei N. Gladkovskii, Nov 18 2011

a(n) = A000819(n) unless n=0.

G.f.: (1/(G(0))-1)/2 where G(k) = 1 - 4*x*(k+1)^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 12 2013

G.f.: T(0)/2 -1/2, where T(k) = 1 - 4*x*(k+1)^2/( 4*x*(k+1)^2 - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013

E.g.f.: sin(x)^2/cos(2*x)= x^2/(1-2*x^2)*T(0), where T(k) = 1 - x^2*(2*k+1)*(2*k+2)/( x^2*(2*k+1)*(2*k+2) + ((k+1)*(2*k+1) - 2*x^2)*((k+2)*(2*k+3) - 2*x^2)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 25 2013

MATHEMATICA

Union[ Range[0, 26]! CoefficientList[ Series[ Sin[x]^2/Cos[ 2x], {x, 0, 26}], x]] (* Robert G. Wilson v, Apr 16 2011 *)

PROG

(PARI) {a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / (2 - 1 / cos(x + x * O(x^m))^2) - 1, m))} /* Michael Somos, Apr 16 2011 */

(Sage)

@CachedFunction

def sp(n, x) :

    if n == 0 : return 1

    return -add(2^(n-k)*sp(k, 1/2)*binomial(n, k) for k in range(n)[::2])

def A000816(n) : return 0 if n == 0 else abs(sp(2*n, x)/2)

[A000816(n) for n in (0..13)]   # Peter Luschny, Jul 30 2012

CROSSREFS

Cf. A000364, A000819, A000822, A000828, A003707, A009125, A009569.

Sequence in context: A292418 A163826 * A000819 A060079 A052502 A209289

Adjacent sequences:  A000813 A000814 A000815 * A000817 A000818 A000819

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 23 13:19 EST 2018. Contains 299581 sequences. (Running on oeis4.)