cp's OEIS Frontend

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

%I A087674 #23 Jun 08 2022 10:11:16
%S A087674 1,1,-1,-3,15,21,-441,477,19935,-101979,-1150281,14838957,60479055,
%T A087674 -2328851979,3529587879,403992301437,-3333935426625,-72778393505979,
%U A087674 1413503392326039,10851976875907917,-554279405351601105,713848745428080021
%N A087674 Value of the n-th Eulerian polynomial (cf. A008292) evaluated at x=-2.
%F A087674 a(n) = -3^(n+1)/2*polylog(-n, -2).
%F A087674 E.g.f.: 3*exp(x)/(2*exp(x)+exp(-2*x)).
%F A087674 O.g.f.: Sum_{n>=0} n! * x^n / Product_{k=1..n} (1 + 3*k*x). [_Paul D. Hanna_, Jul 20 2011]
%F A087674 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
%F A087674 a(n) = (-1)^n * A212846(n). - _Michael Somos_, Aug 27 2018
%F A087674 a(n) = Sum_{k=0..n} Stirling2(n,k) * k! * (-3)^(n-k). - _Ilya Gutkovskiy_, Jun 08 2022
%e A087674 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
%t A087674 Table[-3^(n+1)/2*PolyLog[-n, -2], {n, 0, 21}] (* _Jean-François Alcover_, Apr 26 2013 *)
%t A087674 a[ n_] := If[ n < 0, 0, n! 3/2 SeriesCoefficient[ 1 / (1 + Exp[-3 x] / 2), {x, 0, n}]]; (* _Michael Somos_, Aug 27 2018 *)
%o A087674 (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 */
%o A087674 (PARI) x='x+O('x^66); Vec(serlaplace( 3*exp(x)/(2*exp(x)+exp(-2*x)) )) \\ _Joerg Arndt_, Apr 21 2013
%o A087674 (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 */
%Y A087674 Cf. A000670, A212846.
%K A087674 sign
%O A087674 0,4
%A A087674 _Vladeta Jovovic_, Sep 26 2003
%E A087674 More terms from _Paul D. Hanna_, Jul 20 2011