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A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2. 24

%I

%S 1,-1,5,-61,1385,-50521,2702765,-199360981,19391512145,-2404879675441,

%T 370371188237525,-69348874393137901,15514534163557086905,

%U -4087072509293123892361,1252259641403629865468285,-441543893249023104553682821,177519391579539289436664789665

%N Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.

%C The Euler numbers A000364 with alternating signs.

%C 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 subdiagonal having entries a(i)*C(2*j,2*i) j>=i. - Nolan Wallach (nwallach(AT)ucsd.edu), Dec 26 2005

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

%C 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.

%C To avoid possible confusion: these are the odd e.g.f. coefficients of Gudermannian(x) with the offset shifted by -1 (even coefficients are zero). They are identical to the even e.g.f. coefficients for 1/cosh(x) = -Gudermannian'(x) (see the Example). Since the complex root of cosh(z) with the smallest absolute value is z0 = i*Pi/2, the radius of convergence for the Taylor series of all these functions is Pi/2 = A019669. - _Stanislav Sykora_, Oct 07 2016

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

%H T. D. Noe, <a href="/A028296/b028296.txt">Table of n, a(n) for n = 0..100</a>

%H F. Callegaro, G. Gaiffi, <a href="http://arxiv.org/abs/1406.1304">On models of the braid arrangement and their hidden symmetries</a>, arXiv preprint arXiv:1406.1304 [math.AT], 2014

%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 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 Guodong Liu, <a href="http://dx.doi.org/10.1016/j.jnt.2008.04.003">On congruences of Euler numbers modulo powers of two</a>, Journal of Number Theory, Volume 128, Issue 12, December 2008, Pages 3063-3071.

%H N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen über Differenzenrechnung</a>, Springer 1924, p. 25.

%H Zhi-Hong Sun, <a href="http://arxiv.org/abs/1203.5977">On the further properties of {U_n}</a>, arXiv:1203.5977v1 [math.NT], Mar 27 2012.

%F E.g.f.: sech(x) = 1/cosh(x) (even terms), or Gudermannian(x) (odd terms).

%F Recurrence: a(n) = -sum(i=0..n-1, a(i)*C(2*n, 2*i) ). - _Ralf Stephan_, Feb 24 2005

%F 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

%F a(n) = 2^(4*n+1)*(zeta(-2*n,1/4)-zeta(-2*n,3/4)). - _Gerry Martens_, May 27 2011

%F From _Sergei N. Gladkovskii_, Dec 15 2011 - Oct 09 2013: (START) Continued fractions:

%F 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).

%F 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).

%F 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).

%F G.f.: A(x) = 1/S(0) where S(k) = 1+x*(k+1)^2/S(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^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))).

%F 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)));

%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 G.f.: 1/Q(0), where Q(k) = 1 - sqrt(x) + sqrt(x)*(k+1)/(1-sqrt(x)*(k+1)/Q(k+1));

%F G.f.: (1/Q(0) + 1)/(1-sqrt(x)), where Q(k)= 1 - 1/sqrt(x) + (k+1)*(k+2)/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 (END)

%F n) ~ (-1)^n * (2*n)! * 2^(2*n+2) / Pi^(2*n+1). - _Vaclav Kotesovec_, Aug 04 2014

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

%e Gudermannian(x) = x - 1/6*x^3 + 1/24*x^5 - 61/5040*x^7 + 277/72576*x^9 + ....

%e Gudermannian'(x) = 1/cosh(x) = 1/1!*x^0 - 1/2!*x^2 + 5/4!*x^4 - 61/6!*x^6 + 1385/8!*x^8 + .... - _Stanislav Sykora_, Oct 07 2016

%p 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:

%p seq(A028296(n),n=0..10) ; # _R. J. Mathar_, Apr 20 2011

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

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

%t With[{nn=40},Take[CoefficientList[Series[Gudermannian[x],{x,0,nn}],x] Range[ 0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Feb 24 2018 *)

%o (Maxima)

%o 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 */

%o (Sage)

%o def A028296_list(len):

%o f = lambda k: x*(k+1)^2

%o g = 1

%o for k in range(len-2,-1,-1):

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

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

%o print A028296_list(17)

%o # Alternatively:

%o def A028296(n):

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

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

%o print [A028296(n) for n in (0..16)] # _Peter Luschny_, Aug 10 2015

%o (PARI) a(n) = 2*imag(polylog(-2*n, I)); \\ _Michel Marcus_, May 30 2018

%Y Absolute values are the Euler numbers A000364.

%Y Cf. A019669.

%K sign,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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Last modified November 16 07:34 EST 2018. Contains 317258 sequences. (Running on oeis4.)