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 A028296 Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2. 22
 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 subdiagonal 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. 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 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 [math.AT], 2014 K. Dilcher and C. Vignat, Euler and the Strong Law of Small Numbers, Amer. Math. Mnthly, 123 (May 2016), 486-490. 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) (even terms), or Gudermannian(x) (odd terms). 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 Gudermannian(x) = x - 1/6*x^3 + 1/24*x^5 - 61/5040*x^7 + 277/72576*x^9 + .... 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 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*) 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 *) 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 (PARI) a(n) = 2*imag(polylog(-2*n, I)); \\ Michel Marcus, May 30 2018 CROSSREFS Absolute values are the Euler numbers A000364. Cf. A019669. Sequence in context: A096537 A115047 A000364 * A159316 A231798 A258672 Adjacent sequences:  A028293 A028294 A028295 * A028297 A028298 A028299 KEYWORD sign,easy,nice AUTHOR STATUS approved

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Last modified July 20 15:15 EDT 2018. Contains 312817 sequences. (Running on oeis4.)