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A000831 E.g.f. (1 + tan(x))/(1 - tan(x)). 13
1, 2, 4, 16, 80, 512, 3904, 34816, 354560, 4063232, 51733504, 724566016, 11070525440, 183240753152, 3266330312704, 62382319599616, 1270842139934720, 27507470234550272, 630424777638805504, 15250953398036463616, 388362339077351014400, 10384044045105304174592 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

R. J. Mathar, Table of n, a(n) for n = 0..83

D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.

FORMULA

E.g.f.: tan(x+Pi/4).

a(n)=sum(if even(n+k) ((-1)^((n+k)/2)*sum(j!*stirling2(n,j)*2^(n-j+1)*(-1)^(j)*binomial(j-1,k-1),j,k,n)),k,1,n), n>0. - Vladimir Kruchinin, Aug 19 2010

a(n) = 4^n*(E_{n}(1/2)+E_{n}(1))*(-1)^((n^2-n)/2) for n > 0, where E_{n}(x) is an Euler polynomial. - Peter Luschny, Nov 24 2010

a(n) = 2^n A000111(n). - Gerard P. Michon, Feb 24 2011

E.g.f.: tan(x+Pi/4) = -1 + 2/(1-x*G(0)) ; G(k) = 1 - (x^2)/((x^2) - (2*k + 1)*(2*k + 3)/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 01 2011

E.g.f.: (1 + tan(x))/(1 - tan(x))=1 + 2*x/(U(0)-2*x) where U(k)= 4*k+1 + x/(1+x/(4*k+3 - x/(1- x/U(k+1))));(continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 08 2012

E.g.f.: 1 + 2*x/(G(0)-x) where G(k) =  2*k+1 - x^2/G(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 24 2012

G.f.: 1 + 2*x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - 2*x^2*(2*k+2)*(2*k+3)/Q(k+1) ) ; (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013

E.g.f.: tan(2*x)+sec(2*x)=(x-1)/(x+1) - 2*(2*x^2+3)/( T(0)*3*x*(1+x)- 2*x^2-3)/(x+1), where T(k) = 1 - x^4*(4*k-1)*(4*k+7)/( x^4*(4*k-1)*(4*k+7) - (4*k+1)*(4*k+5)*(16*k^2 + 8*k - 2*x^2 - 3)*(16*k^2 + 40*k - 2*x^2 + 21)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 28 2013

E.g.f.: 1 + 2*x/Q(0), where Q(k) = 4*k+1 -x/(1 - x/( 4*k+3 + x/(1 + x/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2013

E.g.f.: tan(2*x)+sec(2*x)= 2*R(0)-1, where R(k) = 1 + x/( 4*k+1 - x/(1 - x/( 4*k+3 + x/R(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Dec 27 2013

G.f.: 1+ G(0)*2*x/(1-2*x), where G(k) = 1 - 2*x^2*(k+1)*(k+2)/(2*x^2*(k+1)*(k+2) - (1-2*x*(k+1))*(1-2*x*(k+2))/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 24 2014

a(n) ~ n! * (4/Pi)^(n+1). - Vaclav Kotesovec, Jun 15 2015

EXAMPLE

(1+tan x)/(1-tan x)=1+2x/1!+4x^2/2!+16x^3/3!+80x^4/4!+512x^5/5!+...

MAPLE

A000831 := (1+tan(x))/(1-tan(x)) : for n from 0 to 200 do printf("%d %d ", n, n!*coeftayl(A000831, x=0, n)) ; end: # R. J. Mathar, Nov 19 2006

A000831 := n -> `if`(n=0, 1, (-1)^((n^2-n)/2)*4^n*(euler(n, 1/2)+euler( n, 1))): # Peter Luschny, Nov 24 2010

MATHEMATICA

Range[0, 18]! CoefficientList[ Series[(1 + Tan[x])/(1 - Tan[x]), {x, 0, 18}], x] (* Robert G. Wilson v, Apr 16 2011 *)

PROG

(PARI) a(n) = if( n<1, n==0, n! * polcoeff( 1 + 2 / ( 1 / tan( x + x * O(x^n)) - 1), n)) /* Michael Somos, Apr 16 2011 */

(PARI) a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( (cos(x + A) + sin(x + A)) / (cos(x + A) - sin(x + A)), n)) /* Michael Somos, Apr 16 2011 */

(Maxima) a(n):=sum(if evenp(n+k) then ((-1)^((n+k)/2)*sum(j!*stirling2(n, j)*2^(n-j+1)*(-1)^(j)*binomial(j-1, k-1), j, k, n)) else 0, k, 1, n); /* Vladimir Kruchinin, Aug 19 2010 */

(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])

A000831 = lambda n : abs(sp(n, x))

[A000831(n) for n in (0..21)]     # Peter Luschny, Jul 30 2012

CROSSREFS

Bisections: A002436 and A012393.

Cf. A000182, A155100, A258880, A258901, A258994.

Sequence in context: A115125 A025225 A213010 * A000090 A295922 A212432

Adjacent sequences:  A000828 A000829 A000830 * A000832 A000833 A000834

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 18 23:26 EST 2018. Contains 299330 sequences. (Running on oeis4.)