0,3

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).

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]

J. M. Borwein and D. M. Bailey, Mathematics by Experiment, Peters, Boston, 2004; p. 49

J. M. Borwein, P. B. Borwein and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.

G. Chrystal, Algebra, Vol. II, p. 342.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

H. Doerrie, 100 Great Problems of Elementary Mathematics, Dover, NY, 1965, p. 69.

L. Euler, Inst. Calc. Diff., Section 224.

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.

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.

J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.

Knuth, D. E.; Buckholtz, Thomas J. Computation of tangent, Euler and Bernoulli numbers. Math. Comp. 21 1967 663-688.

D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649.

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 444.

D. Shanks, Generalized Euler and class numbers, Math. Comp. 21 (1967), 689-694; 22 (1968), 699.

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).

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.

M. A. Stern, Crelle, 79 (1875), 67-98.

N. J. A. Sloane, The first 100 Euler numbers: Table of n, a(n) for n = 0..99

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.

K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.

D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 144

Michael E. Hoffman, DERIVATIVE POLYNOMIALS, EULER POLYNOMIALS, AND ASSOCIATED INTEGER SEQUENCES

J. Lovejoy and K. Ono, Hypergeometric generating functions for values of Dirichlet and other L-functions, Proc. Nat. Acad. Sci., Vol. 100, No.12, 2003, 6904-6909. [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

Hisanori Mishima, Factorizations of many number sequences

S. Plouffe, The first 7153 Euler numbers (165 megs)

Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

C. Radoux, Determinants de Hankel et theoreme de Sylvester

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, A Famous Application of the Encyclopedia of Integer Sequence (Vugraph from a talk about the OEIS)

R. P. Stanley, Alternating permutations and symmetric functions

Zhi-Wei SUN, Home Page

Sam Wagstaff, Prime divisors of the Bernoulli and Euler numbers

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Alternating Permutation

Wolfram Research, Generating functions for E_n

Index entries for sequences related to boustrophedon transform

Index entries for "core" sequences

E.g.f.: Sum_{n >= 0} a(n)*x^(2*n)/(2*n)! = sec(x) [or gd^(-1)(x)].

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

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]

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

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

a(n) = 2^n * E_n(1/2), where E_n(x) is an Euler polynomial.

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]

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]

a(n) ~ 2^(n+2)*n!/Pi^(n+1) as n -> infinity.

a(n) = Sum_{k = 0..n} A094665(n, k)*2^(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 10 2004

Recurrence: a(n) = -(-1)^n*Sum[i=0..n-1, (-1)^i*a(i)*C(2n, 2i) ]. - Ralf Stephan, Feb 24 2005

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

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

gd^(-1)(x) = log(sec(x) + tan(x)) = log(tan(pi/4 + x/2)).

E.g.f.: Sum_{n >= 0} a(n)*x^(2*n+1)/(2*n+1)! = gd^(-1)(x). - Michael Somos Aug 15 2007

Contribution from Peter Bala (pbala(AT)talktalk.net), Mar 24 2009: (Start)

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.

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)

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)

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.

(End)

sec(x) = 1 + 1/2*x^2 + 5/24*x^4 + 61/720*x^6 + ...

series(sec(x), x, 40): SERIESTOSERIESMULT(%): subs(x=sqrt(y), %): seriestolist(%);

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)

Table[Abs@EulerE[2n], {n, 0, 30}] (from Ray Chandler, Mar 20 2007)

(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)

(PARI) {a(n) = if(n<0, 0, (2*n)! * polcoeff( 1/cos(x + O(x^(2*n+1))), 2*n))}

(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 */

Cf. A000111, A000182, A011248, A060075, A013525, A000816, A002436.

Essentially same as A028296 and A122045.

First column of triangle A060074.

Two main diagonals of triangle A060058 (as iterated sums of squares).

A000464, A002105, A002439, A079144, A158690. [From Peter Bala (pbala(AT)talktalk.net), Mar 24 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jul 06 2009: (Start)

Equals absolute values of row sums of A160485.

(End)

Sequence in context: A096537 A115047 A028296 this_sequence A159316 A116163 A092823

Adjacent sequences: A000361 A000362 A000363 this_sequence A000365 A000366 A000367

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).