A087674 Value of the n-th Eulerian polynomial (cf. A008292) evaluated at x=-2.

1, 1, -1, -3, 15, 21, -441, 477, 19935, -101979, -1150281, 14838957, 60479055, -2328851979, 3529587879, 403992301437, -3333935426625, -72778393505979, 1413503392326039, 10851976875907917, -554279405351601105, 713848745428080021

Offset

0,4

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = -3^(n+1)/2*polylog(-n, -2).

E.g.f.: 3*exp(x)/(2*exp(x)+exp(-2*x)).

O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 3*k*x). [Paul D. Hanna, Jul 20 2011]

G.f.: 1/Q(0), where Q(k)= 1 - x*(k+1)/(1 + x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 20 2013

a(n) = (-1)^n * A212846(n). - Michael Somos, Aug 27 2018

a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * (-3)^(n-k). - Ilya Gutkovskiy, Jun 08 2022

Example

G.f. = 1 + x - x^2 - 3*x^3 + 15*x^4 + 21*x^5 - 441*x^6 + 477*x^7 + 19935*x^8 + ... - Michael Somos, Aug 27 2018

Mathematica

Table[-3^(n+1)/2*PolyLog[-n, -2], {n, 0, 21}] (* Jean-François Alcover, Apr 26 2013 *)
a[ n_] := If[ n < 0, 0, n! 3/2 SeriesCoefficient[ 1 / (1 + Exp[-3 x] / 2), {x, 0, n}]]; (* Michael Somos, Aug 27 2018 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, m!*x^m/prod(k=1, m, 1+3*k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
(PARI) x='x+O('x^66); Vec(serlaplace( 3*exp(x)/(2*exp(x)+exp(-2*x)) )) \\ Joerg Arndt, Apr 21 2013
(PARI) {a(n) = if( n<0, 0, n! * 3/2 * polcoeff( 1 / (1 + exp( -3*x + x * O(x^n)) / 2), n))}; /* Michael Somos, Aug 27 2018 */

Crossrefs

Cf. A000670, A212846.

Sequence in context: A347303 A277585 A318134 * A212846 A276804 A009057

Adjacent sequences: A087671 A087672 A087673 * A087675 A087676 A087677

Keyword

sign

Author

Vladeta Jovovic, Sep 26 2003

Extensions

More terms from Paul D. Hanna, Jul 20 2011

Status

approved