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Triangle read by rows, generalized Eulerian polynomials evaluated at x = -1.
1

%I #10 Oct 07 2020 13:55:26

%S 1,0,1,0,0,-4,0,-2,0,26,0,0,80,352,912,0,16,0,-1936,-11552,-40368,0,0,

%T -3904,-38528,-176832,-560896,-1424960,0,-272,0,297296,3150208,

%U 17187888,65931008,201796240

%N Triangle read by rows, generalized Eulerian polynomials evaluated at x = -1.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized">Generalized Eulerian polynomials</a>.

%F The polynomials are defined P(0,0,x)=1 and P(n,k,x)=(1/2)*Sum_{m=0..n} S(m)*x^m where S(m) = Sum_{j=0..n+1}(-1)^j*binomial(n+1,j)*(k*(m-j)+1)^n*signum(k*(m-j)+1).

%F T(n, k) = P(n, k, -1).

%e Triangle starts:

%e [0] 1

%e [1] 0, 1

%e [2] 0, 0, -4

%e [3] 0, -2, 0, 26

%e [4] 0, 0, 80, 352, 912

%e [5] 0, 16, 0, -1936, -11552, -40368

%e [6] 0, 0, -3904, -38528, -176832, -560896, -1424960

%e [7] 0, -272, 0, 297296, 3150208, 17187888, 65931008, 201796240

%p # The function GeneralizedEulerianPolynomial is defined in A337997.

%p T := (n, k) -> subs(x = -1, GeneralizedEulerianPolynomial(n, k, x)):

%p for n from 0 to 6 do seq(T(n, k), k=0..n) od;

%o (SageMath) # Generalized Eulerian polynomials based on recurrence.

%o @cached_function

%o def EulerianPolynomials(n, k):

%o R.<t> = PolynomialRing(ZZ)

%o if n == 0 or k == 0: return R(k^n)

%o return R((k*t*(1-t)*derivative(EulerianPolynomials(n-1,k), t, 1)

%o + EulerianPolynomials(n-1, k)*(1+(k*n-1)*t)))

%o def T(n, k): return EulerianPolynomials(n, k).substitute(t=-1)

%o for n in (0..7): print([T(n,k) for k in (0..n)])

%Y Cf. A337997, A000182.

%K sign,tabl

%O 0,6

%A _Peter Luschny_, Oct 07 2020