|
|
A028296
|
|
Expansion of e.g.f. Gudermannian(x) = 2*arctan(exp(x)) - Pi/2.
|
|
36
|
|
|
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
|
|
|
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)*binomial(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
Continued fractions:
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).
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).
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).
G.f.: A(x) = 1/S(0) where S(k) = 1 + x*(k+1)^2/S(k+1).
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).
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))).
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)));
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));
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));
G.f.: 1/Q(0), where Q(k) = 1 - sqrt(x) + sqrt(x)*(k+1)/(1-sqrt(x)*(k+1)/Q(k+1));
G.f.: (1/Q(0) + 1)/(1-sqrt(x)), where Q(k) = 1 - 1/sqrt(x) + (k+1)*(k+2)/Q(k+1);
G.f.: Q(0), where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 + 1/Q(k+1)).
(END)
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:
|
|
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 *)
{1, Table[2*(-I)*PolyLog[-2*n, I], {n, 1, 12}]} // Flatten (* Peter Luschny, Aug 12 2021 *)
|
|
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)
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()
(Sage)
shapes = ([x*2 for x in p] for p in Partitions(n))
return sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
(PARI) a(n) = 2*imag(polylog(-2*n, I)); \\ Michel Marcus, May 30 2018
(Python)
from sympy import euler
|
|
CROSSREFS
|
Absolute values are the Euler numbers A000364.
|
|
KEYWORD
|
sign,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|