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

1, 0, 1, 0, 0, -4, 0, -2, 0, 26, 0, 0, 80, 352, 912, 0, 16, 0, -1936, -11552, -40368, 0, 0, -3904, -38528, -176832, -560896, -1424960, 0, -272, 0, 297296, 3150208, 17187888, 65931008, 201796240

Offset: 0

Author

Peter Luschny, Oct 07 2020

Keywords

• sign
• tabl

Examples

			Triangle starts:
[0] 1
[1] 0,    1
[2] 0,    0,    -4
[3] 0,   -2,     0,     26
[4] 0,    0,    80,    352,     912
[5] 0,   16,     0,  -1936,  -11552,   -40368
[6] 0,    0, -3904, -38528, -176832,  -560896, -1424960
[7] 0, -272,     0, 297296, 3150208, 17187888, 65931008, 201796240
		

Links

• Peter Luschny, Generalized Eulerian polynomials.

Crossrefs

Cf. A337997, A000182.

Programs

Maple

# The function GeneralizedEulerianPolynomial is defined in A337997.
T := (n, k) -> subs(x = -1, GeneralizedEulerianPolynomial(n, k, x)):
for n from 0 to 6 do seq(T(n, k), k=0..n) od;

SageMath

# Generalized Eulerian polynomials based on recurrence.
@cached_function
def EulerianPolynomials(n, k):
    R. = PolynomialRing(ZZ)
    if n == 0 or k == 0: return R(k^n)
    return R((k*t*(1-t)*derivative(EulerianPolynomials(n-1,k), t, 1)
           + EulerianPolynomials(n-1, k)*(1+(k*n-1)*t)))
def T(n, k): return EulerianPolynomials(n, k).substitute(t=-1)
for n in (0..7): print([T(n,k) for k in (0..n)])

Formula

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).

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