A337997 Triangle read by rows, generalized Eulerian polynomials evaluated at x = 1.

1, 0, 1, 0, 2, 8, 0, 6, 48, 162, 0, 24, 384, 1944, 6144, 0, 120, 3840, 29160, 122880, 375000, 0, 720, 46080, 524880, 2949120, 11250000, 33592320, 0, 5040, 645120, 11022480, 82575360, 393750000, 1410877440, 4150656720

Offset: 0

Author

Peter Luschny, Oct 07 2020

Keywords

• nonn
• tabl

Examples

			Polynomial triangle starts:
[0] 1
[1] 0, 1
[2] 0, 1+x, x^2+6*x+1
[3] 0, x^2+4*x+1, x^3+23*x^2+23*x+1, 8*x^3+93*x^2+60*x+1
[4] 0, x^3+11*x^2+11*x+1, x^4+76*x^3+230*x^2+76*x+1, 16*x^4+545*x^3+1131*x^2+251*x+
1, 81*x^4+1996*x^3+3446*x^2+620*x+1
Integer triangle starts:
[0] 1
[1] 0,    1
[2] 0,    2,      8
[3] 0,    6,     48,      162
[4] 0,   24,    384,     1944,     6144
[5] 0,  120,   3840,    29160,   122880,    375000
[6] 0,  720,  46080,   524880,  2949120,  11250000,   33592320
[7] 0, 5040, 645120, 11022480, 82575360, 393750000, 1410877440, 4150656720
		

Links

• Peter Luschny, Generalized Eulerian polynomials.

Crossrefs

Cf. A225116, A225117, A225118, A337996, A008292, A060187, A173018, A123125, A284861.

Programs

Maple

# Two alternative implementations are given in the link.
GeneralizedEulerianPolynomial := proc(n, k, x) local S;
   if n = 0 then  return 1 fi;
   S := m -> add((-1)^j*binomial(n+1,j)*(k*(m-j)+1)^n*signum(k*(m-j)+1),j=0..n+1);
   add(S(m)*x^m, m=0..n)/2 end:
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;

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).

T(n, k) = n!*k^n. - Hugo Pfoertner, Oct 07 2020