A156201 Numerator of Euler(n, 1/8).

1, -3, -7, 117, 497, -15123, -95767, 4116837, 34741217, -1921996323, -20273087527, 1370953667157, 17352768515537, -1386843017916723, -20479521468959287, 1888542637550347077, 31872138933891307457, -3331009898404800736323, -63243057486503656319047

Offset

0,2

Links

T. D. Noe, Table of n, a(n) for n = 0..100

Formula

a(n) = Im(2*i*(1+Sum_{j=0..n} (binomial(n,j)*Li_{-j}(i)*4^j))). - Peter Luschny, Apr 29 2013

G.f.: conjecture T(0)/(1+3*x), where T(k) = 1 - 16*x^2*(k+1)^2/(16*x^2*(k+1)^2 + (1+3*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013

a(n) = (-4)^n*skp(n, 3/4), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

a(n) = 2^(4*n+1)*(zeta(-n,1/16)-zeta(-n, 9/16)), where zeta(a, z) is the generalized Riemann zeta function. - Peter Luschny, Mar 11 2015

From Emanuele Munarini, Aug 22 2022: (Start)

E.g.f.: (2*e^t)/(e^(8*t)+1).

E.g.f. for the sequence of the absolute values: (cos(3*t)+sin(3*t))/cos(4*t).

|a(2*n)| = Sum_{k=0..n} binomial(2*n,2*k) (-1)^k 4^(2*n-2*k) 3^(2*k) |E(2*n-2k)|

|a(2*n+1)| = Sum_{k=0..n} binomial(2*n+1,2*k+1) (-1)^k 4^(2*n-2*k) 3^(2*k+1) |E(2*n-2*k)|

where the E(n)'s are the Euler numbers (A122045).

(End)

Maple

p := proc(n) local j; 2*I*(1+add(binomial(n, j)*polylog(-j, I)*4^j, j=0..n)) end:  A156201 := n -> Im(p(n));
seq(A156201(i), i=0..10);  # Peter Luschny, Apr 29 2013

Mathematica

Table[EulerE[n, 1/8] // Numerator, {n, 0, 18}] (* Jean-François Alcover, Apr 30 2013 *)

Crossrefs

For denominators see A001018. Cf. A000813.

Sequence in context: A289629 A015884 A224936 * A066771 A139159 A042329

Adjacent sequences: A156198 A156199 A156200 * A156202 A156203 A156204

Keyword

sign,frac

Author

N. J. A. Sloane, Nov 07 2009

Status

approved