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%I M4019 N1667 #491 Apr 06 2024 09:03:11
%S 1,1,5,61,1385,50521,2702765,199360981,19391512145,2404879675441,
%T 370371188237525,69348874393137901,15514534163557086905,
%U 4087072509293123892361,1252259641403629865468285,441543893249023104553682821,177519391579539289436664789665
%N Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).
%C Inverse Gudermannian gd^(-1)(x) = log(sec(x) + tan(x)) = log(tan(Pi/4 + x/2)) = arctanh(sin(x)) = 2 * arctanh(tan(x/2)) = 2 * arctanh(csc(x) - cot(x)). - _Michael Somos_, Mar 19 2011
%C a(n) is the number of downup permutations of [2n]. Example: a(2)=5 counts 4231, 4132, 3241, 3142, 2143. - _David Callan_, Nov 21 2011
%C a(n) is the number of increasing full binary trees on vertices {0,1,2,...,2n} for which the leftmost leaf is labeled 2n. - _David Callan_, Nov 21 2011
%C a(n) is the number of unordered increasing trees of size 2n+1 with only even degrees allowed and degree-weight generating function given by cosh(t). - _Markus Kuba_, Sep 13 2014
%C a(n) is the number of standard Young tableaux of skew shape (n+1,n,n-1,...,3,2)/(n-1,n-2,...2,1). - _Ran Pan_, Apr 10 2015
%C Since cos(z) has a root at z = Pi/2 and no other root in C with a smaller |z|, the radius of convergence of the e.g.f. (intended complex-valued) is Pi/2 = A019669 (see also A028296). - _Stanislav Sykora_, Oct 07 2016
%C All terms are odd. - _Alois P. Heinz_, Jul 22 2018
%C The sequence starting with a(1) is periodic modulo any odd prime p. The minimal period is (p-1)/2 if p == 1 mod 4 and p-1 if p == 3 mod 4 [Knuth & Buckholtz, 1967, Theorem 2]. - _Allen Stenger_, Aug 03 2020
%C Conjecture: taking the sequence [a(n) : n >= 1] modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, ...] with an apparent period of 6 = phi(21)/2. - _Peter Bala_, May 08 2023
%D 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 J. M. Borwein and D. M. Bailey, Mathematics by Experiment, Peters, Boston, 2004; p. 49
%D J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 141.
%D G. Chrystal, Algebra, Vol. II, p. 342.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.
%D L. Euler, Inst. Calc. Diff., Section 224.
%D S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.
%D L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 269.
%H Seiichi Manyama, <a href="/A000364/b000364.txt">Table of n, a(n) for n = 0..242</a> (terms 0..99 from N. J. A. Sloane)
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H J.-P. Allouche and J. Sondow, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p59">Summation of rational series twisted by strongly B-multiplicative coefficients</a>, Electron. J. Combin., 22 #1 (2015) P1.59; see p. 8.
%H J. L. Arregui, <a href="https://arxiv.org/abs/math/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry3/barry84r2.html">A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2. - From _N. J. A. Sloane_, Dec 27 2012
%H R. Bacher and P. Flajolet, <a href="http://arxiv.org/abs/0901.1379">Pseudo-factorials, elliptic functions, and continued fractions</a>, arXiv:0901.1379 [math.CA], 2009.
%H C. M. Bender and K. A. Milton, <a href="http://arxiv.org/abs/hep-th/9304052">Continued fraction as a discrete nonlinear transform</a>, arXiv:hep-th/9304052, 1993.
%H Natasha Blitvić and Einar Steingrímsson, <a href="https://arxiv.org/abs/2001.00280">Permutations, moments, measures</a>, arXiv:2001.00280 [math.CO], 2020.
%H Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 932.
%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H J. M. Borwein, P. B. Borwein, and K. Dilcher, <a href="http://www.jstor.org/stable/2324715">Pi, Euler numbers and asymptotic expansions</a>, Amer. Math. Monthly, 96 (1989), 681-687.
%H J. M. Borwein and S. T. Chapman, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.03.195">I prefer Pi: A Brief History and Anthology of Articles in the American Mathematical Monthly</a>, Amer. Math. Monthly, 122 (2015), 195-216.
%H R. B. Brent, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Brent/brent5.html">Generalizing Tuenter's Binomial Sums</a>, J. Int. Seq. 18 (2015) # 15.3.2.
%H Richard P. Brent and David Harvey, <a href="http://arxiv.org/abs/1108.0286">Fast computation of Bernoulli, Tangent and Secant numbers</a>, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
%H A. Bucur, J. Lopez-Bonilla, and J. Robles-Garcia, <a href="http://www.bhu.ac.in/journal/vol56-2012/BHU-11.pdf">A note on the Namias identity for Bernoulli numbers</a>, Journal of Scientific Research (Banaras Hindu University, Varanasi), Vol. 56 (2012), 117-120.
%H Elliot J. Carr and Matthew J. Simpson, <a href="https://arxiv.org/abs/1707.06331">Accurate and efficient calculation of response times for groundwater flow</a>, arXiv:1707.06331 [physics.flu-dyn], 2017.
%H Elliot J. Carr and Matthew J. Simpson, <a href="https://arxiv.org/abs/1810.08890">New homogenization approaches for stochastic transport through heterogeneous media</a>, arXiv:1810.08890 [physics.bio-ph], 2018.
%H K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/CHEN/AlgBE2.html">Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer Sequences, 4 (2001), #01.1.6.
%H Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2212.07232">Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers</a>, arXiv:2212.07232 [math.CO], 2022.
%H Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2304.06545">Continued fractions for cycle-alternating permutations</a>, arXiv:2304.06545 [math.CO], 2023.
%H K. Dilcher and C. Vignat, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.123.5.486">Euler and the Strong Law of Small Numbers</a>, Amer. Math. Mnthly, 123 (May 2016), 486-490.
%H Filippo Disanto and Emanuele Munarini, <a href="https://doi.org/10.1007/s42452-019-0564-7">Local height in weighted Dyck models of random walks and the variability of the number of coalescent histories for caterpillar-shaped gene trees and species trees</a>, SN Applied Sciences (2019), 1:578.
%H D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/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 A. L. Edmonds and S, Klee, <a href="http://arxiv.org/abs/1210.7396">The combinatorics of hyperbolized manifolds</a>, arXiv preprint arXiv:1210.7396 [math.CO], 2012. - From _N. J. A. Sloane_, Jan 02 2013
%H C. J. Fewster and D. Siemssen, <a href="http://arxiv.org/abs/1403.1723">Enumerating Permutations by their Run Structure</a>, arXiv preprint arXiv:1403.1723 [math.CO], 2014.
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 144.
%H D. Foata and M.-P. Schutzenberger, <a href="http://www-igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1973NombresEulerPermAlternantesColorado.pdf">Nombres d'Euler et permutations alternantes</a>, in J. N. Srivastava et al., eds., A Survey of Combinatorial Theory (North Holland Publishing Company, Amsterdam, 1973), pp. 173-187.
%H Dominique Foata and Guo-Niu Han, <a href="http://irma.math.unistra.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
%H J. M. Hammersley, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/14_1_1.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23.
%H Michael E. Hoffman, <a href="https://doi.org/10.37236/1453">Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences</a>, The Electronic Journal of Combinatorics, vol.6, no.1, #R21, (1999).
%H Donald E. Knuth and Thomas J. Buckholtz, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0221735-9">Computation of tangent, Euler and Bernoulli numbers</a>, Math. Comp. 21 1967 663-688.
%H D. E. Knuth and Thomas J. Buckholtz <a href="/A000182/a000182.pdf"> Computation of tangent, Euler and Bernoulli numbers</a>, Math. Comp. 21 1967 663-688. [Annotated scanned copy]
%H D. H. Lehmer, <a href="http://www.jstor.org/stable/1968647">Lacunary recurrence formulas for the numbers of Bernoulli and Euler</a>, Annals Math., 36 (1935), 637-649.
%H Guodong Liu, <a href="http://www.fq.math.ca/Papers1/43-2/paper43-2-7.pdf">On Congruences of Euler Numbers Modulo an Odd Square</a>, Fib. Q., 43,2 (2005), 132-136.
%H 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, Mar 24 2009]
%H F. Luca and P. Stanica, <a href="http://calhoun.nps.edu/bitstream/handle/10945/29605/LucaStanicaJCNTfinal.pdf?sequence=1">On some conjectures on the monotonicity of some arithmetical sequences</a>, J. Combin. Number Theory 4 (2012) 1-10.
%H P. Luschny, <a href="http://www.luschny.de/math/primes/eulerinc.html">Approximation, inclusion and asymptotics of the Euler numbers.</a>
%H J. Malenfant, <a href="http://arxiv.org/abs/1103.1585">Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers</a>, arxiv:1103.1585 [math.NT], 2011.
%H Miguel Méndez and Rafael Sánchez, <a href="https://arxiv.org/abs/1707.00336">On the combinatorics of Riordan arrays and Sheffer polynomials: monoids, operads and monops</a>, arXiv:1707.00336 [math.CO], 2017, Section 4.3, Example 4.
%H Miguel A. Méndez and Rafael Sánchez Lamoneda, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p25">Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials</a>, The Electronic Journal of Combinatorics 25(3) (2018), #P3.25.
%H R. Mestrovic, <a href="http://arxiv.org/abs/1212.3602">A search for primes p such that Euler number E_{p-3} is divisible by p</a>, arXiv preprint arXiv:1212.3602 [math.NT], 2012. - From _N. J. A. Sloane_, Jan 25 2013
%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha133.htm">Factorizations of Euler numbers n=0..78</a>, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1331.htm">n=80..106</a>.
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Munarini/muna25.html">Two-Parameter Identities for q-Appell Polynomials</a>, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]
%H Simon Plouffe, <a href="http://plouffe.fr/simon/OEIS/b000364.txt.gz">68000 terms, up to E(34000)</a> (2.1 gigas)
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H C. Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
%H Paolo Emilio Ricci, <a href="https://doi.org/10.3390/axioms8020050">Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey</a>, Axioms (2019) Vol. 8, No. 2, 50.
%H D. Shanks, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0223295-5">Generalized Euler and class numbers</a>. Math. Comp. 21 (1967) 689-694.
%H D. Shanks, <a href="/A000003/a000003.pdf">Generalized Euler and class numbers</a>, Math. Comp. 21 (1967), 689-694; 22 (1968), 699. [Annotated scanned copy]
%H Vladimir Shevelev, <a href="http://www.emis.de/journals/INTEGERS/papers/m1/m1.Abstract.html">The number of permutations with prescribed up-down structure as a function of two variables</a>, INTEGERS, 12 (2012), #A1. - From _N. J. A. Sloane_, Feb 07 2013
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%H N. J. A. Sloane, <a href="/A000364/a000364.jpg">A Famous Application of the Encyclopedia of Integer Sequence</a> (Vugraph from a talk about the OEIS)
%H R. P. Stanley, <a href="https://arxiv.org/abs/math/0603520">Alternating permutations and symmetric functions</a>, arXiv:math/0603520 [math.CO], 2006.
%H R. P. Stanley, <a href="http://jointmathematicsmeetings.org/meetings/national/jmm/colloq-10.pdf">Permutations</a>
%H M. A. Stern, <a href="http://dx.doi.org/10.1515/crll.1880.89.257">Zur Theorie der Eulerschen Zahlen</a>, J. Reine Angew. Math., 79 (1875), 67-98.
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2005.01.001">On Euler numbers modulo powers of two</a>, Journal of Number Theory, Volume 115, Issue 2, December 2005, Pages 371-380.
%H D. C. Vella, <a href="http://www.emis.de/journals/INTEGERS/papers/i1/i1.Abstract.html">Explicit Formulas for Bernoulli and Euler Numbers</a>, Integers 8(1), A1, 2008.
%H Sam Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/full.pdf">Prime divisors of the Bernoulli and Euler numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerNumber.html">Euler Number</a>, <a href="http://mathworld.wolfram.com/SecantNumber.html">Secant Number</a>, <a href="http://mathworld.wolfram.com/AlternatingPermutation.html">Alternating Permutation</a>.
%H Wolfram Research, <a href="http://functions.wolfram.com/IntegerFunctions/EulerE/11">Generating functions for E_n</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F E.g.f.: Sum_{n >= 0} a(n) * x^(2*n) / (2*n)! = sec(x). - _Michael Somos_, Aug 15 2007
%F E.g.f.: Sum_{n >= 0} a(n) * x^(2*n+1) / (2*n+1)! = gd^(-1)(x). - _Michael Somos_, Aug 15 2007
%F 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 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 Also, for positive odd n, log(2) - Sum_{k = 1..(n-1)/2} (-1)^(k-1)/k ~ (-1)^((n-1)/2) * Sum_{k >= 0} (-1)^k*E(k)/n^(2*k+1), where E(k) is the k-th Euler number, by Borwein, Borwein, and Dilcher, Lemma 2 with f(x) := 1/(x + 1/2), h := 1/2 and then replace x with (n-1)/2. - _Peter Bala_, Oct 29 2016
%F 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 Deléham_, Sep 04 2005
%F 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 a(n) = 2^n * E_n(1/2), where E_n(x) is an Euler polynomial.
%F a(k) = a(j) (mod 2^n) if and only if k == j (mod 2^n) (k and j are even). [Stern; see also Wagstaff and Sun]
%F 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 a(n) ~ 2^(2*n+2)*(2*n)!/Pi^(2*n+1) as n -> infinity. [corrected by _Vaclav Kotesovec_, Jul 10 2021]
%F a(n) = Sum_{k=0..n} A094665(n, k)*2^(n-k). - _Philippe Deléham_, Jun 10 2004
%F Recurrence: a(n) = -(-1)^n*Sum_{i=0..n-1} (-1)^i*a(i)*binomial(2*n, 2*i). - _Ralf Stephan_, Feb 24 2005
%F O.g.f.: 1/(1-x/(1-4*x/(1-9*x/(1-16*x/(...-n^2*x/(1-...)))))) (continued fraction due to T. J. Stieltjes). - _Paul D. Hanna_, Oct 07 2005
%F a(n) = (Integral_{t=0..Pi} log(tan(t/2)^2)^(2n)dt)/Pi^(2n+1). - Logan Kleinwaks (kleinwaks(AT)alumni.princeton.edu), Mar 15 2007
%F From _Peter Bala_, Mar 24 2009: (Start)
%F Basic hypergeometric generating function: 2*exp(-t)*Sum {n >= 0} 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 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 Sum_{n>=0} a(n)*z^(2*n)/(4*n)!! = Beta(1/2-z/(2*Pi),1/2+z/(2*Pi))/Beta(1/2,1/2) with Beta(z,w) the Beta function. - _Johannes W. Meijer_, Jul 06 2009
%F a(n) = Sum_(Sum_(binomial(k,m)*(-1)^(n+k)/(2^(m-1))*Sum_(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n), n>0. - _Vladimir Kruchinin_, Aug 05 2010
%F If n is prime, then a(n)==1 (mod 2*n). - _Vladimir Shevelev_, Sep 04 2010
%F From _Peter Bala_, Jan 21 2011: (Start)
%F (1)... a(n) = (-1/4)^n*B(2*n,-1),
%F where {B(n,x)}n>=1 = [1, 1+x, 1+6*x+x^2, 1+23*x+23*x^2+x^3, ...] is the sequence of Eulerian polynomials of type B - see A060187. Equivalently,
%F (2)... a(n) = Sum_{k = 0..2*n} Sum_{j = 0..k} (-1)^(n-j) *binomial(2*n+1,k-j)*(j+1/2)^(2*n).
%F We also have
%F (3)... a(n) = 2*A(2*n,i)/(1+i)^(2*n+1),
%F where i = sqrt(-1) and where {A(n,x)}n>=1 = [x, x + x^2, x + 4*x^2 + x^3, ...] denotes the sequence of Eulerian polynomials - see A008292. Equivalently,
%F (4)... a(n) = i*Sum_{k = 1..2*n} (-1)^(n+k)*k!*Stirling2(2*n,k) *((1+i)/2)^(k-1)
%F = i*Sum_{k = 1..2*n} (-1)^(n+k)*((1+i)/2)^(k-1) Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*j^(2*n).
%F Either this explicit formula for a(n) or (2) above may be used to obtain congruence results for a(n). For example, for prime p
%F (5a)... a(p) = 1 (mod p)
%F (5b)... a(2*p) = 5 (mod p)
%F and for odd prime p
%F (6a)... a((p+1)/2) = (-1)^((p-1)/2) (mod p)
%F (6b)... a((p-1)/2) = -1 + (-1)^((p-1)/2) (mod p).
%F (End)
%F a(n) = (-1)^n*2^(4*n+1)*(zeta(-2*n,1/4) - zeta(-2*n,3/4)). - _Gerry Martens_, May 27 2011
%F a(n) may be expressed as a sum of multinomials taken over all compositions of 2*n into even parts (Vella 2008): a(n) = Sum_{compositions 2*i_1 + ... + 2*i_k = 2*n} (-1)^(n+k)* multinomial(2*n, 2*i_1, ..., 2*i_k). For example, there are 4 compositions of the number 6 into even parts, namely 6, 4+2, 2+4 and 2+2+2, and hence a(3) = 6!/6! - 6!/(4!*2!) - 6!/(2!*4!) + 6!/(2!*2!*2!) = 61. A companion formula expressing a(n) as a sum of multinomials taken over the compositions of 2*n-1 into odd parts has been given by Malenfant 2011. - _Peter Bala_, Jul 07 2011
%F a(n) = the upper left term in M^n, where M is an infinite square production matrix; M[i,j] = A000290(i) = i^2, i >= 1 and 1 <= j <= i+1, and M[i,j] = 0, i >= 1 and j >= i+2 (see examples). - _Gary W. Adamson_, Jul 18 2011
%F E.g.f. A'(x) satisfies the differential equation A'(x)=cos(A(x)). - _Vladimir Kruchinin_, Nov 03 2011
%F From _Peter Bala_, Nov 28 2011: (Start)
%F a(n) = D^(2*n)(cosh(x)) evaluated at x = 0, where D is the operator cosh(x)*d/dx. a(n) = D^(2*n-1)(f(x)) evaluated at x = 0, where f(x) = 1+x+x^2/2! and D is the operator f(x)*d/dx.
%F Other generating functions: cosh(Integral_{t = 0..x} 1/cos(t)) dt = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + .... Cf. A012131.
%F A(x) := arcsinh(tan(x)) = log( sec(x) + tan(x) ) = x + x^3/3! + 5*x^5/5! + 61*x^7/7! + 1385*x^9/9! + .... A(x) satisfies A'(x) = cosh(A(x)).
%F B(x) := Series reversion( log(sec(x) + tan(x)) ) = x - x^3/3! + 5*x^5/5! - 61*x^7/7! + 1385*x^9/9! - ... = arctan(sinh(x)). B(x) satisfies B'(x) = cos(B(x)). (End)
%F HANKEL transform is A097476. PSUM transform is A173226. - _Michael Somos_, May 12 2012
%F a(n+1) - a(n) = A006212(2*n). - _Michael Somos_, May 12 2012
%F a(0) = 1 and, for n > 0, a(n) = (-1)^n*((4*n+1)/(2*n+1) - Sum_{k = 1..n} (4^(2*k)/2*k)*binomial(2*n,2*k-1)*A000367(k)/A002445(k)); see the Bucur et al. link. - _L. Edson Jeffery_, Sep 17 2012
%F O.g.f.: Sum_{n>=0} (2*n)!/2^n * x^n / Product_{k=1..n} (1 + k^2*x). - _Paul D. Hanna_, Sep 20 2012
%F From _Sergei N. Gladkovskii_, Oct 31 2011 to Oct 11 2013: (Start)
%F Continued fractions:
%F E.g.f.: (sec(x)) = 1+x^2/T(0), T(k) = 2(k+1)(2k+1) - x^2 + x^2*(2k+1)(2k+2)/T(k+1).
%F E.g.f.: 2/Q(0) where Q(k) = 1 + 1/(1 - x^2/(x^2 - 2*(k+1)*(2*k+1)/Q(k+1))).
%F G.f.: 1/Q(0) where Q(k) = 1 + x*k*(3*k-1) - x*(k+1)*(2*k+1)*(x*k^2+1)/Q(k+1).
%F E.g.f.: (2 + x^2 + 2*U(0))/(2 + (2 - x^2)*U(0)) where U(k)= 4*k + 4 + 1/( 1 + x^2/(2 - x^2 + (2*k+3)*(2*k+4)/U(k+1))).
%F E.g.f.: 1/cos(x) = 8*(x^2+1)/(4*x^2 + 8 - x^4*U(0)) where U(k) = 1 + 4*(k+1)*(k+2)/(2*k+3 - x^2*(2*k+3)/(x^2 - 8*(k+1)*(k+2)*(k+3)/U(k+1))).
%F G.f.: 1/U(0) where U(k) = 1 + x - x*(2*k+1)*(2*k+2)/(1 - x*(2*k+1)*(2*k+2)/U(k+1)).
%F G.f.: 1 + x/G(0) where G(k) = 1 + x - x*(2*k+2)*(2*k+3)/(1 - x*(2*k+2)*(2*k+3)/G(k+1)).
%F Let F(x) = sec(x^(1/2)) = Sum_{n>=0} a(n)*x^n/(2*n)!, then F(x)=2/(Q(0) + 1) where Q(k)= 1 - x/(2*k+1)/(2*k+2)/(1 - 1/(1 + 1/Q(k+1))).
%F G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/Q(k+1)).
%F E.g.f.: 1/cos(x) = 1 + x^2/(2-x^2)*Q(0), where Q(k) = 1 - 2*x^2*(k+1)*(2*k+1)/( 2*x^2*(k+1)*(2*k+1)+ (12-x^2 + 14*k + 4*k^2)*(2-x^2 + 6*k + 4*k^2)/Q(k+1)). (End)
%F a(n) = Sum_{k=1..2*n} (Sum_{i=0..k-1} (i-k)^(2*n)*binomial(2*k,i)*(-1)^(i+k+n)) / 2^(k-1) for n>0, a(0)=1. - _Vladimir Kruchinin_, Oct 05 2012
%F It appears that a(n) = 3*A076552(n -1) + 2*(-1)^n for n >= 1. Conjectural congruences: a(2*n) == 5 (mod 60) for n >= 1 and a(2*n+1) == 1 (mod 60) for n >= 0. - _Peter Bala_, Jul 26 2013
%F From _Peter Bala_, Mar 09 2015: (Start)
%F O.g.f.: Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 - sqrt(-x)*(2*k + 1)) = Sum_{n >= 0} 1/2^n * Sum_{k = 0..n} (-1)^k*binomial(n,k)/(1 + x*(2*k + 1)^2).
%F O.g.f. is 1 + x*d/dx(log(F(x))), where F(x) = 1 + x + 3*x^2 + 23*x^3 + 371*x^4 + ... is the o.g.f. for A255881. (End)
%F Sum_(n >= 1, A034947(n)/n^(2d+1)) = a(d)*Pi^(2d+1)/(2^(2d+2)-2)(2d)! for d >= 0; see Allouche and Sondow, 2015. - _Jonathan Sondow_, Mar 21 2015
%F Asymptotic expansion: 4*(4*n/(Pi*e))^(2*n+1/2)*exp(1/2+1/(24*n)-1/(2880*n^3) +1/(40320*n^5)-...). (See the Luschny link.) - _Peter Luschny_, Jul 14 2015
%F a(n) = 2*(-1)^n*Im(Li_{-2n}(i)), where Li_n(x) is polylogarithm, i=sqrt(-1). - _Vladimir Reshetnikov_, Oct 22 2015
%F Limit_{n->infinity} ((2*n)!/a(n))^(1/(2*n)) = Pi/2. - _Stanislav Sykora_, Oct 07 2016
%F O.g.f.: 1/(1 + x - 2*x/(1 - 2*x/(1 + x - 12*x/(1 - 12*x/(1 + x - 30*x/(1 - 30*x/(1 + x - ... - (2*n - 1)*(2*n)*x/(1 - (2*n - 1)*(2*n)*x/(1 + x - ... ))))))))). - _Peter Bala_, Nov 09 2017
%F For n>0, a(n) = (-PolyGamma(2*n, 1/4) / 2^(2*n - 1) - (2^(2*n + 2) - 2) * Gamma(2*n + 1) * zeta(2*n + 1)) / Pi^(2*n + 1). - _Vaclav Kotesovec_, May 04 2020
%F a(n) ~ 2^(4*n + 3) * n^(2*n + 1/2) / (Pi^(2*n + 1/2) * exp(2*n)) * exp(Sum_{k>=1} bernoulli(k+1) / (k*(k+1)*2^k*n^k)). - _Vaclav Kotesovec_, Mar 05 2021
%F Peter Bala's conjectured congruences, a(2n) == 5 (mod 60) for n >= 1 and a(2n + 1) == 1 (mod 60), hold due to the results of Stern (mod 4) and Knuth & Buckholtz (mod 3 and 5). - _Charles R Greathouse IV_, Mar 23 2022
%e G.f. = 1 + x + 5*x^2 + 61*x^3 + 1385*x^4 + 50521*x^5 + 2702765*x^6 + 199360981*x^7 + ...
%e sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + ...
%e From _Gary W. Adamson_, Jul 18 2011: (Start)
%e The first few rows of matrix M are:
%e 1, 1, 0, 0, 0, ...
%e 4, 4, 4, 0, 0, ...
%e 9, 9, 9, 9, 0, ...
%e 16, 16, 16, 16, 16, ... (End)
%p series(sec(x),x,40): SERIESTOSERIESMULT(%): subs(x=sqrt(y),%): seriestolist(%);
%p # end of program
%p A000364_list := proc(n) local S,k,j; S[0] := 1;
%p for k from 1 to n do S[k] := k*S[k-1] od;
%p for k from 1 to n do
%p for j from k to n do
%p S[j] := (j-k)*S[j-1]+(j-k+1)*S[j] od od;
%p seq(S[j], j=1..n) end:
%p A000364_list(16); # _Peter Luschny_, Apr 02 2012
%p A000364 := proc(n)
%p abs(euler(2*n)) ;
%p end proc: # _R. J. Mathar_, Mar 14 2013
%t Take[ Range[0, 32]! * CoefficientList[ Series[ Sec[x], {x, 0, 32}], x], {1, 32, 2}] (* _Robert G. Wilson v_, Apr 23 2006 *)
%t Table[Abs[EulerE[2n]], {n, 0, 30}] (* _Ray Chandler_, Mar 20 2007 *)
%t a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Sec[ x], {x, 0, m}]]]; (* _Michael Somos_, Nov 22 2013 *)
%t a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, m! SeriesCoefficient[ InverseGudermannian[ x], {x, 0, m}]]]; (* _Michael Somos_, Nov 22 2013 *)
%t a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]* (2j-m)^(2n), {j, 0, m/2}] (-1)^(k-m), {m, 0, k}], {k, 1, 2n}]; Table[ a[n], {n, 0, 16}] (* _Jean-François Alcover_, Jun 26 2019, after _Vladimir Kruchinin_ *)
%o (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]))} \\ _Paul D. Hanna_ Oct 07 2005
%o (PARI) {a(n) = if( n<0, 0, (2*n)! * polcoeff( 1 / cos(x + O(x^(2*n + 1))), 2*n))}; /* _Michael Somos_, Jun 18 2002 */
%o (PARI) {a(n) = my(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 */
%o (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!/2^m * x^m/prod(k=1, m, 1+k^2*x+x*O(x^n))), n)} \\ _Paul D. Hanna_, Sep 20 2012
%o (PARI) list(n)=my(v=Vec(1/cos(x+O(x^(2*n+1)))));vector(n,i,v[2*i-1]*(2*i-2)!) \\ _Charles R Greathouse IV_, Oct 16 2012
%o (PARI) a(n)=subst(bernpol(2*n+1),'x,1/4)*4^(2*n+1)*(-1)^(n+1)/(2*n+1) \\ _Charles R Greathouse IV_, Dec 10 2014
%o (PARI) a(n)=abs(eulerfrac(2*n)) \\ _Charles R Greathouse IV_, Mar 23 2022
%o (Maxima) a(n):=sum(sum(binomial(k,m)*(-1)^(n+k)/(2^(m-1))*sum(binomial(m,j)*(2*j-m)^(2*n),j,0,m/2)*(-1)^(k-m),m,0,k),k,1,2*n); /* _Vladimir Kruchinin_, Aug 05 2010 */
%o (Maxima) a[n]:=if n=0 then 1 else sum(sum((i-k)^(2*n)*binomial(2*k, i)*(-1)^(i+k+n), i, 0, k-1)/ (2^(k-1)), k, 1, 2*n); makelist(a[n], n, 0, 16); \\ _Vladimir Kruchinin_, Oct 05 2012
%o (Sage)
%o # Algorithm of L. Seidel (1877)
%o # n -> [a(0), a(1), ..., a(n-1)] for n > 0.
%o def A000364_list(len) :
%o R = []; A = {-1:0, 0:1}; k = 0; e = 1
%o for i in (0..2*len-1) :
%o Am = 0; A[k + e] = 0; e = -e
%o for j in (0..i) : Am += A[k]; A[k] = Am; k += e
%o if e < 0 : R.append(A[-i//2])
%o return R
%o A000364_list(17) # _Peter Luschny_, Mar 31 2012
%o (Python)
%o from functools import lru_cache
%o from math import comb
%o @lru_cache(maxsize=None)
%o def A000364(n): return 1 if n == 0 else (1 if n % 2 else -1)*sum((-1 if i % 2 else 1)*A000364(i)*comb(2*n,2*i) for i in range(n)) # _Chai Wah Wu_, Jan 14 2022
%Y Cf. A000111, A000182, A011248, A019669, A034947, A060075, A013525, A000816, A002436, A000464, A002105, A002439, A079144, A158690.
%Y Essentially same as A028296 and A122045.
%Y First column of triangle A060074.
%Y Two main diagonals of triangle A060058 (as iterated sums of squares).
%Y Absolute values of row sums of A160485. - _Johannes W. Meijer_, Jul 06 2009
%Y Left edge of triangle A210108, see also A125053, A076552. Cf. A255881.
%Y Bisection (even part) of A317139.
%Y The sequences [(-k^2)^n*Euler(2*n, 1/k), n = 0, 1, ...] are: A000007 (k=1), A000364 (k=2), |A210657| (k=3), A000281 (k=4), A272158 (k=5), A002438 (k=6), A273031 (k=7).
%K nonn,easy,nice,core,changed
%O 0,3
%A _N. J. A. Sloane_
%E Typo in name corrected by _Anders Claesson_, Dec 01 2015
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