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A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2. 20
1, -1, 5, -61, 1385, -50521, 2702765, -199360981, 19391512145, -2404879675441, 370371188237525, -69348874393137901, 15514534163557086905, -4087072509293123892361, 1252259641403629865468285, -441543893249023104553682821, 177519391579539289436664789665 (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) = c*Sum_{k=1..n} binomial(2n,2k)*a(n-k). For c = -3, -2, -1, 1, 2, 3, 4 this is A210676, A210657, A028296, A094088, A210672, A210674, A249939.

REFERENCES

K. Dilcher and C. Vignat, Euler and the Strong Law of Small Numbers, Amer. Math. Mnthly, 123 (May 2016), 486-490.

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 [math.AT], 2014

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

Guodong Liu, On congruences of Euler numbers modulo powers of two, Journal of Number Theory, Volume 128, Issue 12, December 2008, Pages 3063-3071.

N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 25.

Zhi-Hong Sun, On the further properties of {U_n}, arXiv:1203.5977v1 [math.NT], 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

a(n) = 2*Im(Li_{-2n}(i)), where Li_n(x) is polylogarithm, i=sqrt(-1). - Vladimir Reshetnikov, Oct 22 2015

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 */

(Sage)

def A028296_list(len):

    f = lambda k: x*(k+1)^2

    g = 1

    for k in range(len-2, -1, -1):

        g = (1-f(k)/(f(k)+1/g)).simplify_rational()

    return taylor(g, x, 0, len-1).list()

print A028296_list(17)

# Alternatively:

def A028296(n):

    shapes = [map(lambda x: x*2, p) for p in Partitions(n).list()]

    return sum([(-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes])

print [A028296(n) for n in (0..16)] # Peter Luschny, Aug 10 2015

CROSSREFS

Absolute values are the Euler numbers A000364.

Sequence in context: A096537 A115047 A000364 * A159316 A231798 A258672

Adjacent sequences:  A028293 A028294 A028295 * A028297 A028298 A028299

KEYWORD

sign,easy,nice,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 31 17:28 EDT 2016. Contains 273548 sequences.