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
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.
Programs
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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;
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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).