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

%I A151919 #35 May 27 2024 07:15:32
%S A151919 1,-4,34,-442,7654,-165634,4301254,-130313362,4512058774,
%T A151919 -175757170114,7606919927974,-362157366660082,18809374928573494,
%U A151919 -1058311485335621794,64126470727596628294,-4163172358878650459602,288297029592971540217814,-21212159439736738874060674
%N A151919 a(n) = (-2)^n*A_{n,3}(1/2) where A_{n,k}(x) are the generalized Eulerian polynomials.
%C A151919 Old name was: row sums in A154594.
%H A151919 G. C. Greubel, <a href="/A151919/b151919.txt">Table of n, a(n) for n = 0..360</a>
%H A151919 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized">Generalized Eulerian polynomials</a>.
%F A151919 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
%F A151919 a(n) = (-1)^n*Sum_{k=0..n} binomial(n,k)*3^k*A000670(k). - _Emanuele Munarini_, Dec 05 2020
%t A151919 m = 18; CoefficientList[Exp[-x]/(2 - Exp[-3x]) + O[x]^m, x]*Range[0, m-1]! (* _Jean-François Alcover_, Jun 19 2019 *)
%o A151919 (SageMath)
%o A151919 @CachedFunction
%o A151919 def BB(n, k, x):  # modified cardinal B-splines
%o A151919     if n == 1: return 0 if (x < 0) or (x >= k) else 1
%o A151919     return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
%o A151919 def EulerianPolynomial(n, k, x):
%o A151919     if n == 0: return 1
%o A151919     return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
%o A151919 [(-2)^n*EulerianPolynomial(n, 3, 1/2) for n in (0..17)]
%o A151919 # _Peter Luschny_, May 04 2013
%o A151919 (Magma)
%o A151919 R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(-x)/(2 - Exp(-3*x)) ))); // _G. C. Greubel_, May 27 2024
%Y A151919 Cf. A154594 (row sums), A284861 (row sums if unsigned).
%Y A151919 Cf. A000670 (Fubini numbers).
%K A151919 sign
%O A151919 0,2
%A A151919 _Roger L. Bagula_, Jan 12 2009
%E A151919 New name and more terms added by _Peter Luschny_, May 04 2013