This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002436 M3701 N1512 #92 Apr 15 2025 08:27:38
%S A002436 1,4,80,3904,354560,51733504,11070525440,3266330312704,
%T A002436 1270842139934720,630424777638805504,388362339077351014400,
%U A002436 290870261262635870715904,260290690801376575335956480,274278793184290987427604987904,336150887870579862992197737512960
%N A002436 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(2*x).
%D A002436 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.
%D A002436 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.
%D A002436 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002436 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002436 Alois P. Heinz, <a href="/A002436/b002436.txt">Table of n, a(n) for n = 0..216</a>
%F A002436 a(n) = A000831(2*n) = 4^n * A000364(n). a(n) = 2 * A000816(n) except n=0. - _Michael Somos_, Apr 26 2011
%F A002436 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
%F A002436 E.g.f.: sec(2*x) = 1/cos(2*x) = 1/(cos(x)^2 - sin(x)^2). - _Arkadiusz Wesolowski_, Jul 25 2012
%F A002436 From _Sergei N. Gladkovskii_, Oct 23 2012 (Start)
%F A002436 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).
%F A002436 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)
%F A002436 G.f.: 1/U(0) where U(k) = 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
%F A002436 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
%F A002436 a(n+1) = | 2*16^n*lerchphi(-1, -2*n, 1/2) |, n>=0. - _Peter Luschny_, Apr 27 2013
%F A002436 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
%F A002436 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
%F A002436 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
%e A002436 G.f. = 1 + 4*x + 80*x^2 + 3904*x^3 + 354560*x^4 + 51733504*x^5 + 11070525440*x^6 + ...
%p A002436 A := n -> (-4)^n*euler(2*n); # (Then A(n) = a(n+1) for n >= 0.) # _Peter Luschny_, Jan 27 2009
%t A002436 Rest@ Union[ Range[0, 24]! CoefficientList[ Series[ Sec[ 2x], {x, 0, 24}], x]] (* _Robert G. Wilson v_, Apr 16 2011 *)
%t A002436 a[ n_] := If[ n < 0, 0, 2 (-16)^n LerchPhi[ -1, -2 n, 1/2]]; (* _Michael Somos_, Oct 14 2014 *)
%t A002436 With[{nn=30},Take[CoefficientList[Series[Sec[2x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, May 06 2018 *)
%o A002436 (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 */
%o A002436 (Sage)
%o A002436 @CachedFunction
%o A002436 def sp(n,x) :
%o A002436 if n == 0 : return 1
%o A002436 return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
%o A002436 A002436 = lambda n : abs(sp(2*(n-1),x))
%o A002436 [A002436(n) for n in (1..15)] # _Peter Luschny_, Jul 30 2012
%o A002436 (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
%Y A002436 Cf. A000364, A000816, A000831.
%Y A002436 (-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
%K A002436 nonn,easy,nice
%O A002436 0,2
%A A002436 _N. J. A. Sloane_
%E A002436 More terms from _Michael Somos_, Jun 21 2002