|
| |
|
|
A002436
|
|
E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x).
(Formerly M3701 N1512)
|
|
18
|
|
|
|
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.
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
|
|
|
|
FORMULA
|
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
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))
(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
|
(-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
|
|
|
KEYWORD
|
nonn,easy,nice
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
