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A337996 Triangle read by rows, generalized Eulerian polynomials evaluated at x = -1.

%I A337996 #10 Oct 07 2020 13:55:26
%S A337996 1,0,1,0,0,-4,0,-2,0,26,0,0,80,352,912,0,16,0,-1936,-11552,-40368,0,0,
%T A337996 -3904,-38528,-176832,-560896,-1424960,0,-272,0,297296,3150208,
%U A337996 17187888,65931008,201796240
%N A337996 Triangle read by rows, generalized Eulerian polynomials evaluated at x = -1.
%H A337996 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized">Generalized Eulerian polynomials</a>.
%F A337996 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 A337996 T(n, k) = P(n, k, -1).
%e A337996 Triangle starts:
%e A337996 [0] 1
%e A337996 [1] 0,    1
%e A337996 [2] 0,    0,    -4
%e A337996 [3] 0,   -2,     0,     26
%e A337996 [4] 0,    0,    80,    352,     912
%e A337996 [5] 0,   16,     0,  -1936,  -11552,   -40368
%e A337996 [6] 0,    0, -3904, -38528, -176832,  -560896, -1424960
%e A337996 [7] 0, -272,     0, 297296, 3150208, 17187888, 65931008, 201796240
%p A337996 # The function GeneralizedEulerianPolynomial is defined in A337997.
%p A337996 T := (n, k) -> subs(x = -1, GeneralizedEulerianPolynomial(n, k, x)):
%p A337996 for n from 0 to 6 do seq(T(n, k), k=0..n) od;
%o A337996 (SageMath) # Generalized Eulerian polynomials based on recurrence.
%o A337996 @cached_function
%o A337996 def EulerianPolynomials(n, k):
%o A337996     R.<t> = PolynomialRing(ZZ)
%o A337996     if n == 0 or k == 0: return R(k^n)
%o A337996     return R((k*t*(1-t)*derivative(EulerianPolynomials(n-1,k), t, 1)
%o A337996            + EulerianPolynomials(n-1, k)*(1+(k*n-1)*t)))
%o A337996 def T(n, k): return EulerianPolynomials(n, k).substitute(t=-1)
%o A337996 for n in (0..7): print([T(n,k) for k in (0..n)])
%Y A337996 Cf. A337997, A000182.
%K A337996 sign,tabl
%O A337996 0,6
%A A337996 _Peter Luschny_, Oct 07 2020