A151919 a(n) = (-2)^n*A_{n,3}(1/2) where A_{n,k}(x) are the generalized Eulerian polynomials.

1, -4, 34, -442, 7654, -165634, 4301254, -130313362, 4512058774, -175757170114, 7606919927974, -362157366660082, 18809374928573494, -1058311485335621794, 64126470727596628294, -4163172358878650459602, 288297029592971540217814, -21212159439736738874060674

Offset: 0

Author

Roger L. Bagula, Jan 12 2009

Keywords

• sign

Comments

Old name was: row sums in A154594.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..360
• Peter Luschny, Generalized Eulerian polynomials.

Crossrefs

Cf. A154594 (row sums), A284861 (row sums if unsigned).

Cf. A000670 (Fubini numbers).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(-x)/(2 - Exp(-3*x)) ))); // G. C. Greubel, May 27 2024

Mathematica

m = 18; CoefficientList[Exp[-x]/(2 - Exp[-3x]) + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Jun 19 2019 *)

SageMath

@CachedFunction
def BB(n, k, x):  # modified cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= k) else 1
    return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
    if n == 0: return 1
    return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
[(-2)^n*EulerianPolynomial(n, 3, 1/2) for n in (0..17)]
# Peter Luschny, May 04 2013

Formula

E.g.f.: exp(-x)/(2 - exp(-3*x)). (see e.g.f. of row sums of A284861 with x -> -x). - Wolfdieter Lang, Jul 12 2017

a(n) = (-1)^n*Sum_{k=0..n} binomial(n,k)*3^k*A000670(k). - Emanuele Munarini, Dec 05 2020

Extensions

New name and more terms added by Peter Luschny, May 04 2013