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A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.

%I A179929 #41 Apr 17 2025 14:55:54
%S A179929 1,2,2,-6,-30,42,882,954,-39870,-203958,2300562,29677914,-120958110,
%T A179929 -4657703958,-7059175758,807984602874,6667870853250,-145556787011958,
%U A179929 -2827006784652078,21703953751815834,1108558810703202210
%N A179929 a(n) = 2^n*A(n, -1/2), A(n, x) the Eulerian polynomials.
%H A179929 Stanislav Sykora, <a href="/A179929/b179929.txt">Table of n, a(n) for n = 0..199</a>
%H A179929 OEIS Wiki, <a href="http://oeis.org/wiki/Eulerian_polynomials">Eulerian polynomials</a>
%F A179929 From _Philippe Deléham_, Dec 22 2011: (Start)
%F A179929 a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k
%F A179929 a(n) = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. (End)
%F A179929 From _Peter Bala_, Mar 12 2013: (Start)
%F A179929 E.g.f.: 3/(1 + 2exp(-3x)) = 1 + 2x + 2x^2/2! - 6x^3/3! - 30x^4/4! + ....
%F A179929 Recurrence equation: a(n+1) = 3a(n) - Sum_{k=0..n} binomial(n,k) a(k)a(n-k).
%F A179929 (-1)^n*a(n) are the coefficients of a delta operator associated with a sequence of polynomials of binomial type - see A195205. (End)
%F A179929 a(n) ~ n! * 2*3^(n+1)/(Pi^2+(log(2))^2)^(n/2+1) * (Pi*sin(n*arctan(Pi/log(2))) - log(2)*cos(n*arctan(Pi/log(2)))). - _Vaclav Kotesovec_, Oct 09 2013
%F A179929 From _Stanislav Sykora_, May 15 2014: (Start)
%F A179929 a(n) = -2*A212846(n) for n > 0.
%F A179929 a(n) = -3^(n+1)*Li(-n, -1/2), with Li(-n, x) = Sum_{k>=0} ((k^n)*(x^k)) the polylogarithm.
%F A179929 a(n) = Sum_{k=0..n} 3^(n-k)*(-1)^k*k!*S(n+1, k+1), S(m, l) the Stirling number of second kind. (End)
%t A179929 a[n_] := Sum[3^(n-k) (-1)^k k! StirlingS2[n+1, k+1], {k, 0, n}];
%t A179929 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 17 2019 *)
%o A179929 (PARI) A179929(n) = {my(s,k,term);
%o A179929   term = 3^n;s = term*stirling(n+1,1,2);
%o A179929   for (k=1,n,term *= -k/3;s += term*stirling(n+1,k+1,2););
%o A179929 return(s);} \\ _Stanislav Sykora_, May 15 2014
%Y A179929 Cf. A000629 = 2^n*A(n, 1/2).
%Y A179929 Cf. A212846, A123125, A173018, A195205.
%K A179929 sign
%O A179929 0,2
%A A179929 _Peter Luschny_, Aug 03 2010