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.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B45.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
Beáta Bényi and Toshiki Matsusaka, Extensions of the combinatorics of poly-Bernoulli numbers, arXiv:2106.05585 [math.CO], 2021.
F. Callegaro and 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.
Emanuele Munarini, Two-Parameter Identities for q-Appell Polynomials, Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.1.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 25.
Joe Santmyer, Derivative Polynomials for Trigonometric and Hyperbolic Functions, Arhimede Math. J. (2023) Vol. 10, No. 2, 152-158.
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)*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
From Sergei N. Gladkovskii, Dec 15 2011 - Oct 09 2013: (Start)
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) ~ (-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 *)
{1, Table[2*(-I)*PolyLog[-2*n, I], {n, 1, 12}]} // Flatten (* Peter Luschny, Aug 12 2021 *)
a[0] := 1; a[n_] := a[n] = -Sum[Binomial[2 n, 2 k] a[k], {k, 0, n - 1}]; Map[a, Range[0, 16]] (* Oliver Seipel, May 19 2024 *)
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))
(Sage)
def A028296(n):
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)
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
(PARI) a(n)=eulerfrac(2*n) \\ Charles R Greathouse IV, Mar 23 2022
(Python)
from sympy import euler
def A028296(n): return euler(n<<1) # Chai Wah Wu, Apr 16 2023
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
STATUS
approved