%I A000364 M4019 N1667
%S A000364 1,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,370371188237525,
%T A000364 69348874393137901,15514534163557086905,4087072509293123892361,
%U A000364 1252259641403629865468285,441543893249023104553682821,177519391579539289436664789665,
               80723299235887898062168247453281
%N A000364 Euler (or secant or "Zig") numbers: expansion of sec x.
%D A000364 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 810; gives a version with signs: E_{2n} = (-1)^n*a(n) 
               (this is A028296).
%D A000364 R. BACHER AND P. FLAJOLET, PSEUDO-FACTORIALS, ELLIPTIC FUNCTIONS AND 
               CONTINUED FRACTIONS, arXiv 0901.1379. [Added by N. J. A. Sloane (njas(AT)research.att.com), 
               Feb 01 2009]
%D A000364 J. M. Borwein and D. M. Bailey, Mathematics by Experiment, Peters, Boston, 
               2004; p. 49
%D A000364 J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic 
               expansions, Amer. Math. Monthly, 96 (1989), 681-687.
%D A000364 G. Chrystal, Algebra, Vol. II, p. 342.
%D A000364 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A000364 H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 
               1965, p. 69.
%D A000364 L. Euler, Inst. Calc. Diff., Section 224.
%D A000364 D. Foata and M.-P. Schutzenberger, Nombres d'Euler et permutations alternantes, 
               in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory 
               (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
%D A000364 Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, 
               Eulerian, MacMahon and Stirling number triangles, Journal of Integer 
               Sequences, Vol. 9 (2006), Article 06.4.1.
%D A000364 J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 
               14 (1989), 1-23.
%D A000364 Knuth, D. E.; Buckholtz, Thomas J. Computation of tangent, Euler and 
               Bernoulli numbers. Math. Comp. 21 1967 663-688.
%D A000364 D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli 
               and Euler, Annals Math., 36 (1935), 637-649.
%D A000364 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; 
               see p. 444.
%D A000364 D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 
               689-694; 22 (1968), 699.
%D A000364 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000364 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000364 Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the 
               Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 
               06.1.1.
%D A000364 M. A. Stern, Crelle, 79 (1875), 67-98.
%H A000364 N. J. A. Sloane, <a href="b000364.txt">The first 100 Euler numbers: Table 
               of n, a(n) for n = 0..99</a>
%H A000364 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A000364 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A000364 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer 
               Sequences, 4 (2001), #01.1.6.
%H A000364 D. Dumont and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/
               paper/publication.html">Polynomes d'Euler et les fractions continues 
               de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%H A000364 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/
               Publications/books.html">Analytic Combinatorics</a>, 2009; see page 
               144
%H A000364 Michael E. Hoffman, <a href="http://www.emis.ams.org/journals/EJC/Volume_6/
               PDF/v6i1r21.pdf">DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED 
               INTEGER SEQUENCES</a>
%H A000364 J. Lovejoy and K. Ono, <a href="http://www.pnas.org/content/100/12/6904.abstract?ck=nck">
               Hypergeometric generating functions for values of Dirichlet and other 
               L-functions</a>, Proc. Nat. Acad. Sci., Vol. 100, No.12, 2003, 6904-6909. 
               [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha133.htm">Factorizations of many number sequences</a>
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha1331.htm">Factorizations of many number sequences</a>
%H A000364 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A000364 S. Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/OEIS/b000364.txt">
               The first 7153 Euler numbers</a> (165 megs)
%H A000364 Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/
               index.html">Arithmetic and growth of periodic orbits</a>, J. Integer 
               Seqs., Vol. 4 (2001), #01.2.1.
%H A000364 C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">
               Determinants de Hankel et theoreme de Sylvester</a>
%H A000364 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/doc/sg.txt">
               My favorite integer sequences</a>, in Sequences and their Applications 
               (Proceedings of SETA '98).
%H A000364 N. J. A. Sloane, <a href="a000364.jpg">A Famous Application of the Encyclopedia 
               of Integer Sequence</a> (Vugraph from a talk about the OEIS)
%H A000364 R. P. Stanley, <a href="http://arXiv.org/abs/math.CO/0603520">Alternating 
               permutations and symmetric functions</a>
%H A000364 Zhi-Wei SUN, <a href="http://pweb.nju.edu.cn/zwsun">Home Page</a>
%H A000364 Sam Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/
               full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>
%H A000364 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EulerNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000364 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               SecantNumber.html">Link to a section of The World of Mathematics.</
               a>
%H A000364 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               AlternatingPermutation.html">Alternating Permutation</a>
%H A000364 Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/
               EulerE/11">Generating functions for E_n</a>
%H A000364 <a href="Sindx_Bo.html#boustrophedon">Index entries for sequences related 
               to boustrophedon transform</a>
%H A000364 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(x) [or gd^(-1)(x)].
%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = 2*arctanh(cosec(x)-cotan(x)). 
               - Ralf Stephan, Dec 16 2004
%F A000364 Pi/4 - [Sum_{k=0..n-1} (-1)^k/(2*k+1)] ~ (1/2)*[Sum_{k>=0} (-1)^k*E(k)/
               (2*n)^(2k+1)] for positive even n. [Borwein, Borwein and . Dilcher]
%F A000364 Let M_n be the n X n matrix M_n(i, j) = binomial(2*i, 2*(j-1)) = A086645(i, 
               j-1); then for n>0, a(n) = det(M_n); example : det([1, 1, 0, 0; 1, 
               6, 1, 0; 1, 15, 15, 1; 1, 28, 70, 28 ]) = 1385 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Sep 04 2005
%F A000364 This sequence is also (-1)^n EulerE[2 n] or Abs[EulerE[2 n]]. - Paul 
               Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006
%F A000364 a(n) = 2^n * E_n(1/2), where E_n(x) is an Euler polynomial.
%F A000364 a(k)=a(l) (mod 2^n) if and only if k=l (mod 2^n) (k and l are even). 
               [Stern; see also Wagstaff and Sun]
%F A000364 E_k(3^{k+1}+1)/4=(3^k/2)Sum_{j=0}^{2^n-1}(-1)^{j-1}(2j+1)^k[(3j+1)/2^n] 
               (mod 2^n) where k is even and [x] is the greatest integer function. 
               [Sun]
%F A000364 a(n) ~ 2^(n+2)*n!/Pi^(n+1) as n -> infinity.
%F A000364 a(n) = Sum_{k = 0..n} A094665(n, k)*2^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jun 10 2004
%F A000364 Recurrence: a(n) = -(-1)^n*Sum[i=0..n-1, (-1)^i*a(i)*C(2n, 2i) ]. - Ralf 
               Stephan, Feb 24 2005
%F A000364 O.g.f.: A(x) = 1/(1-x/(1-4*x/(1-9*x/(1-16*x/(...-n^2*x/(1-...)))))) (continued 
               fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2005
%F A000364 a(n)=Integrate[Log[Tan[t/2]^2]^(2n),{t,0,Pi}]/Pi^(2n+1). - Logan Kleinwaks 
               (kleinwaks(AT)alumni.princeton.edu), Mar 15 2007
%F A000364 gd^(-1)(x) = log(sec(x) + tan(x)) = log(tan(pi/4 + x/2)).
%F A000364 E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = gd^(-1)(x). - Michael 
               Somos Aug 15 2007
%F A000364 Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 24 2009: (Start)
%F A000364 Basic hypergeometric generating function: 2*exp(-t)*Sum {n = 0..inf} 
               Product {k = 1..n} (1-exp(-(4*k-2)*t))*exp(-2*n*t)/Product {k = 1..n+1} 
               (1+exp(-(4*k-2)*t)) = 1 + t + 5*t^2/2! + 61*t^3/3! + .... For other 
               sequences with generating functions of a similar type see A000464, 
               A002105, A002439, A079144 and A158690.
%F A000364 a(n) = 2*(-1)^n*L(-2*n), where L(s) is the Dirichlet L-function L(s) 
               = 1 - 1/3^s + 1/5^s - + .... (End)
%F A000364 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%F A000364 sum(a(n)*z^(2*n)/(4*n)!!, n=0..infinity) = Beta(1/2-z/(2*Pi),1/2+z/(2*Pi))/
               Beta(1/2,1/2) with Beta(z,w) the Beta function.
%F A000364 (End)
%e A000364 sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + ...
%p A000364 series(sec(x),x,40): SERIESTOSERIESMULT(%): subs(x=sqrt(y),%): seriestolist(%);
%t A000364 Take[ Range[0, 32]!*CoefficientList[ Series[ Sec[x], {x, 0, 32}], x], 
               {1, 32, 2}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 23 2006)
%t A000364 Table[Abs@EulerE[2n], {n, 0, 30}] (from Ray Chandler, Mar 20 2007)
%o A000364 (PARI) {a(n)=local(CF=1+x*O(x^n));if(n<0,return(0), for(k=1,n,CF=1/(1-(n-k+1)^2*x*CF));
               return(Vec(CF)[n+1]))} (Hanna)
%o A000364 (PARI) {a(n) = if(n<0, 0, (2*n)! * polcoeff( 1/cos(x + O(x^(2*n+1))), 
               2*n))}
%o A000364 (PARI) {a(n) = local(A); if(n<0, 0, n = 2*n+1 ; A = x*O(x^n); n! * polcoeff( 
               log(1/cos(x+A) + tan(x+A)), n))} /* Michael Somos Aug 15 2007 */
%Y A000364 Cf. A000111, A000182, A011248, A060075, A013525, A000816, A002436.
%Y A000364 Essentially same as A028296 and A122045.
%Y A000364 First column of triangle A060074.
%Y A000364 Two main diagonals of triangle A060058 (as iterated sums of squares).
%Y A000364 A000464, A002105, A002439, A079144, A158690. [From Peter Bala (pbala(AT)talktalk.net), 
               Mar 24 2009]
%Y A000364 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 
               2009: (Start)
%Y A000364 Equals absolute values of row sums of A160485.
%Y A000364 (End)
%Y A000364 Sequence in context: A096537 A115047 A028296 this_sequence A159316 A116163 
               A092823
%Y A000364 Adjacent sequences: A000361 A000362 A000363 this_sequence A000365 A000366 
               A000367
%K A000364 nonn,easy,nice,core
%O A000364 0,3
%A A000364 N. J. A. Sloane (njas(AT)research.att.com).