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A002436 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x).
(Formerly M3701 N1512)
10
1, 4, 80, 3904, 354560, 51733504, 11070525440, 3266330312704, 1270842139934720, 630424777638805504, 388362339077351014400, 290870261262635870715904, 260290690801376575335956480, 274278793184290987427604987904, 336150887870579862992197737512960 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 75.

Glaisher, J. W. L., Quart. J. Pure Appl. Math., 44 (1913), 105-112.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..100

FORMULA

a(n) = A000831(2*n) = 4^n * A000364(n). a(n) = 2 * A000816(n) except n=0. - Michael Somos, Apr 26 2011

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

E.g.f.: sec(2*x) = 1/cos(2*x) = 1/(cos(x)^2 - sin(x)^2). - Arkadiusz Wesolowski, Jul 25 2012

From Sergei N. Gladkovskii, Oct 23 2012 (Start)

G.f.: 1/U(0) where U(k)=  1 - 2*(4*k+1)*(4*k+2)*x/( 1 - 2*(4*k+3)*(4*k+4)*x/U(k+1)); (continued fraction, 2-step).

E.g.f.: 1/S(0) where S(k)= 1 - 2*x^2/((4*k+1)*(2*k+1) - x^2*(4*k+1)*(2*k+1)/(x^2 - (4*k+3)*(k+1)/S(k+1); (continued fraction, 3rd kind, 3-step). (End)

G.f.: 1/U(0) where U(k,x)= 1 - (4*k+2)*(4*k+2)*x^2/(1 - (4*k+4)*(4*k+4)*x^2/U(k+1); (continued fraction, 2-step). - Sergei N. Gladkovskii, Nov 06 2012

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

a(n+1) = | 2*16^n*lerchphi(-1, -2*n, 1/2) |, n>=0. - Peter Luschny, Apr 27 2013

G.f.: Q(0), where Q(k) = 1 - x*(2*k+2)^2/( x*(2*k+2)^2 - 1/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013

E.g.f.: sec(2*x) = 1/cos(2*x)= 1+ 2*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

a(n) = (-1)^n*2^(6*n+1)*(Zeta(-2*n,1/4) - Zeta(-2*n, 3/4)), where Zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015

EXAMPLE

G.f. = 1 + 4*x + 80*x^2 + 3904*x^3 + 354560*x^4 + 51733504*x^5 + 11070525440*x^6 + ...

MAPLE

A := n -> (-4)^n*euler(2*n); (Then A(n) = a(n+1) for n >= 0.) # Peter Luschny, Jan 27 2009

MATHEMATICA

Rest@ Union[ Range[0, 24]! CoefficientList[ Series[ Sec[ 2x], {x, 0, 24}], x]] (* Robert G. Wilson v, Apr 16 2011 *)

a[ n_] :=  If[ n < 0, 0, 2 (-16)^n LerchPhi[ -1, -2 n, 1/2]]; (* Michael Somos, Oct 14 2014 *)

PROG

(PARI) {a(n) = local(m); if( n<0, 0, m = 2*n; m! * polcoeff( 1 / cos( 2*x + x * O(x^m)), 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])

A002436 = lambda n : abs(sp(2*(n-1), x))

[A002436(n) for n in (1..15)]   # Peter Luschny, Jul 30 2012

CROSSREFS

Cf. A000364, A000816, A000831.

Sequence in context: A012057 A214298 A102063 * A013031 A012923 A012922

Adjacent sequences:  A002433 A002434 A002435 * A002437 A002438 A002439

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Michael Somos, Jun 21, 2002

STATUS

approved

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Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.