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A028296 E.g.f. Gudermannian(x) = 2*arctan(exp(x))-Pi/2. 17
1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145, -2404879675441, 370371188237525, -69348874393137901, 15514534163557086905, -4087072509293123892361, 1252259641403629865468285 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The Euler numbers A000364 with alternating signs.

The first column of the inverse to the matrix with entries C(2*i,2*j), i,j >=0. The full matrix is lower triangular with the i-th sundiagonal having entries a(i)*C(2*j,2*i) j>=i. - Nolan Wallach (nwallach(AT)ucsd.edu), Dec 26 2005

This sequence is also EulerE(2*n). - Paul Abbott (paul(AT)physics.uwa.edu.au), Apr 14 2006

Consider the sequence defined by a(0)=1; thereafter a(n) = 2*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For k = -3, -2, -1, 1, 2, 3 this is A210676, A210657, A028296, A094088, A210672, A210674.

REFERENCES

Gradshteyn and Ryzhik, Tables, 5th ed., Section 1.490, pp. 51-52.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

F. Callegaro, G. Gaiffi, On models of the braid arrangement and their hidden symmetries, arXiv preprint arXiv:1406.1304, 2014

A. L. Edmonds and S, Klee, The combinatorics of hyperbolized manifolds, arXiv preprint arXiv:1210.7396, 2012. - From N. J. A. Sloane, Jan 02 2013

N. E. Noerlund, Vorlesungen ueber Differenzenrechnung, Springer 1924, p. 25.

Zhi-Hong Sun, On the further properties of {U_n}, arXiv:1203.5977v1, Mar 27 2012.

FORMULA

E.g.f.: sech(x) = 1/cosh(x), or gd(x).

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

a(n) = sum_{k=1,3,5,..,2n+1} (-1)^((k-1)/2) /(2^k*k) *sum_{i=0..k} (-1)^i*(k-2*i)^(2n+1) *binomial(k,i); [Vladimir Kruchinin, Apr 20 2011]

a(n) = 2^(4*n+1)*(zeta(-2*n,1/4)-zeta(-2*n,3/4)) - [Gerry Martens, May 27 2011]

G.f.: A(x) = 1 - x/(S(0)+x), S(k) = euler(2*k) + x*euler(2*k+2) - x*euler(2*k)*euler(2*k+4)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011

E.g.f.: E(x) = 1 - x/(S(0)+x); S(k) = (k+1)*euler(2*k) + x*euler(2*k+2) - x*(k+1)*euler(2*k)*euler(2*k+4)/S(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011

2*arctan(exp(z))-Pi/2 = z*(1 - z^2/(G(0) + z^2)), G(k) = 2*(k+1)*(2*k+3)*euler(2*k) + z^2*euler(2*k+2) - 2*z^2*(k+1)*(2*k+3)*euler(2*k)*euler(2*k+4)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011

G.f.: A(x) = 1/S(0) where S(k) = 1+x*(k+1)^2/S(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jun 22 2012

From Sergei N. Gladkovskii, Sep 27 2012 (Start)

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) ; (continued fraction, Euler's 1st kind, 1-step).

E.g.f.:(2 - x^4/( (x^2+2)*Q(0) + 2))/(2+x^2) where Q(k)=  4*k + 4 + 1/( 1 - x^2/( 2 + x^2 + (2*k+3)*(2*k+4)/Q(k+1))); (continued fraction, Euler's 1st kind, 3-step).

(End)

E.g.f.: 1/cosh(x)=8*(1-x^2)/(8 - 4*x^2 - 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))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 30 2012

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)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 15 2012

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)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 16 2012

G.f.: 1/Q(0), where Q(k) = 1 - sqrt(x) + sqrt(x)*(k+1)/(1-sqrt(x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 21 2013

G.f.: (1/Q(0) + 1)/(1-sqrt(x)), where Q(k)= 1 - 1/sqrt(x) + (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 26 2013

G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 + 1/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013

a(n) ~ (-1)^n * (2*n)! * 2^(2*n+2) / Pi^(2*n+1). - Vaclav Kotesovec, Aug 04 2014

EXAMPLE

gd x = x - 1/6*x^3 + 1/24*x^5 - 61/5040*x^7 + 277/72576*x^9 + ....

MAPLE

A028296 := proc(n) a :=0 ; for k from 1 to 2*n+1 by 2 do a := a+(-1)^((k-1)/2)/2^k/k *add( (-1)^i *(k-2*i)^(2*n+1) *binomial(k, i), i=0..k) ; end do: a ; end proc:

seq(A028296(n), n=0..10) ; # R. J. Mathar, Apr 20 2011

MATHEMATICA

Table[EulerE[2*n], {n, 0, 30}] (* Paul Abbott, Apr 14 2006 *)

Table[(CoefficientList[Series[1/Cosh[x], {x, 0, 40}], x]*Range[0, 40]!)[[2*n+1]], {n, 0, 20}] (* Vaclav Kotesovec, Aug 04 2014*)

PROG

(Maxima)

a(n):=sum((-1+(-1)^(k))*(-1)^((k+1)/2)/(2^(k+1)*k)*sum((-1)^i*(k-2*i)^n*binomial(k, i), i, 0, k), k, 1, n); /* with interpolated zeros, Vladimir Kruchinin, Apr 20 2011 */

CROSSREFS

Absolute values are the Euler numbers A000364.

Sequence in context: A196125 A096537 A115047 * A000364 A159316 A231798

Adjacent sequences:  A028293 A028294 A028295 * A028297 A028298 A028299

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 22 08:25 EDT 2014. Contains 248388 sequences.