login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A225147 a(n) = Im((1-I)^(1-n)*A_{n, 3}(I)) where A_{n, k}(x) are the generalized Eulerian polynomials. 1
-1, 2, 5, -46, -205, 3362, 22265, -515086, -4544185, 135274562, 1491632525, -54276473326, -718181418565, 30884386347362, 476768795646785, -23657073914466766, -417370516232719345, 23471059057478981762, 465849831125196593045, -29279357851856595135406 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..19.

Peter Luschny, Generalized Eulerian polynomials.

FORMULA

a(n) = Im(-2*I*(1+sum_{j=0..n}(binomial(n,j)*Li{-j}(I)*3^j))).

For a recurrence see the Maple program.

G.f.: conjecture -T(0)/(1+2*x), where T(k) = 1 - 9*x^2*(k+1)^2/(9*x^2*(k+1)^2 + (1+2*x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 12 2013

a(n) = -(-3)^n*skp(n, 2/3), where skp(n,x) are the Swiss-Knife polynomials A153641. - Peter Luschny, Apr 19 2014

G.f.: A225147 = -1/T(0), where T(k) = 1 + 2*x + (k+1)^2*(3*x)^2/ T(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 29 2014

E.g.f.: -exp(-2*x)*sech(3*x). - Sergei N. Gladkovskii, Sep 29 2014

a(n) ~ n! * (sqrt(3)*sin(Pi*n/2) - cos(Pi*n/2)) * 2^(n+1) * 3^n / Pi^(n+1). - Vaclav Kotesovec, Sep 29 2014

From Peter Bala, Nov 13 2016: (Start)

a(n) = - 6^n*E(n,1/6), where E(n,x) denotes the Euler polynomial of order n.

a(2*n) = (-1)^(n+1)*A002438(n); a(2*n+1) = 1/2*(-1)^n*A002439(n). (End)

MAPLE

B := proc(n, u, k) option remember;

if n = 1 then if (u < 0) or (u >= 1) then 0 else 1 fi

else k*u*B(n-1, u, k) + k*(n-u)*B(n-1, u-1, k) fi end:

EulerianPolynomial := proc(n, k, x) local m; if x = 0 then RETURN(1) fi;

add(B(n+1, m+1/k, k)*u^m, m = 0..n); subs(u=x, %) end:

seq(Im((1-I)^(1-n)*EulerianPolynomial(n, 3, I)), n=0..19);

MATHEMATICA

CoefficientList[Series[-E^(-2*x)*Sech[3*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 29 2014 after Sergei N. Gladkovskii *)

Table[-6^n EulerE[n, 1/6], {n, 0, 19}] (* Peter Luschny, Nov 16 2016 after Peter Bala *)

PROG

(Sage)

from mpmath import *

mp.dps = 32; mp.pretty = True

def A225147(n): return im(-2*I*(1+add(binomial(n, j)*polylog(-j, I)*3^j for j in (0..n))))

[A225147(n) for n in (0..19)]

CROSSREFS

Cf. A000810 (real part (up to sign)), A212435 (k=2), A122045 (k=1), A002438, A002439.

Sequence in context: A056680 A005166 A121621 * A119715 A023273 A041729

Adjacent sequences:  A225144 A225145 A225146 * A225148 A225149 A225150

KEYWORD

sign,easy

AUTHOR

Peter Luschny, Apr 30 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 30 20:18 EDT 2017. Contains 284302 sequences.