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

1, 2, 2, -6, -30, 42, 882, 954, -39870, -203958, 2300562, 29677914, -120958110, -4657703958, -7059175758, 807984602874, 6667870853250, -145556787011958, -2827006784652078, 21703953751815834, 1108558810703202210

Offset: 0

Author

Peter Luschny, Aug 03 2010

Keywords

• sign

Links

• Stanislav Sykora, Table of n, a(n) for n = 0..199
• OEIS Wiki, Eulerian polynomials

Crossrefs

Cf. A000629 = 2^n*A(n, 1/2).

Cf. A212846, A123125, A173018, A195205.

Programs

Mathematica

a[n_] := Sum[3^(n-k) (-1)^k k! StirlingS2[n+1, k+1], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 17 2019 *)

PARI

A179929(n) = {my(s,k,term);
  term = 3^n;s = term*stirling(n+1,1,2);
  for (k=1,n,term *= -k/3;s += term*stirling(n+1,k+1,2););
return(s);} \\ Stanislav Sykora, May 15 2014

Formula

From Philippe Deléham, Dec 22 2011: (Start)

a(n) = Sum_{k, 0<=k<=n} A123125(n,k)*(-1)^(n-k)*2^k

a(n) = Sum_{k, 0<=k<=n} A173018(n,k)*2^(n-k)*(-1)^k. (End)

From Peter Bala, Mar 12 2013: (Start)

E.g.f.: 3/(1 + 2exp(-3x)) = 1 + 2x + 2x^2/2! - 6x^3/3! - 30x^4/4! + ....

Recurrence equation: a(n+1) = 3a(n) - Sum_{k=0..n} binomial(n,k) a(k)a(n-k).

(-1)^n*a(n) are the coefficients of a delta operator associated with a sequence of polynomials of binomial type - see A195205. (End)

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

From Stanislav Sykora, May 15 2014: (Start)

a(n) = -2*A212846(n) for n > 0.

a(n) = -3^(n+1)*Li(-n, -1/2), with Li(-n, x) = Sum_{k>=0} ((k^n)*(x^k)) the polylogarithm.

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)