A002436 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x). (Formerly M3701 N1512)

1, 4, 80, 3904, 354560, 51733504, 11070525440, 3266330312704, 1270842139934720, 630424777638805504, 388362339077351014400, 290870261262635870715904, 260290690801376575335956480, 274278793184290987427604987904, 336150887870579862992197737512960

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.

J. W. L. Glaisher, On the last two figures in certain coefficients analogous to the Eulerian numbers, 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 = 0..216

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 *)
With[{nn=30}, Take[CoefficientList[Series[Sec[2x], {x, 0, nn}], x] Range[0, nn]!, {1, -1, 2}]] (* Harvey P. Dale, May 06 2018 *)

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
(Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1+Tan(x))/(1-Tan(x)) )); [Factorial(n-1)*b[n]: n in [1..m by 2]]; // Vincenzo Librandi, May 30 2019

Crossrefs

Cf. A000364, A000816, A000831.

(-1)^n*a(n) give the alternating row sums of A060187(2*n), n >= 0. The alternating sums for odd numbered rows vanish. - Wolfdieter Lang, Jul 12 2017

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