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

%I A337997 #18 Oct 07 2020 18:18:42
%S A337997 1,0,1,0,2,8,0,6,48,162,0,24,384,1944,6144,0,120,3840,29160,122880,
%T A337997 375000,0,720,46080,524880,2949120,11250000,33592320,0,5040,645120,
%U A337997 11022480,82575360,393750000,1410877440,4150656720
%N A337997 Triangle read by rows, generalized Eulerian polynomials evaluated at x = 1.
%H A337997 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/EulerianPolynomialsGeneralized">Generalized Eulerian polynomials</a>.
%F A337997 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 A337997 T(n, k) = P(n, k, 1).
%F A337997 T(n, k) = n!*k^n. - _Hugo Pfoertner_, Oct 07 2020
%e A337997 Polynomial triangle starts:
%e A337997 [0] 1
%e A337997 [1] 0, 1
%e A337997 [2] 0, 1+x, x^2+6*x+1
%e A337997 [3] 0, x^2+4*x+1, x^3+23*x^2+23*x+1, 8*x^3+93*x^2+60*x+1
%e A337997 [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+
%e A337997 1, 81*x^4+1996*x^3+3446*x^2+620*x+1
%e A337997 Integer triangle starts:
%e A337997 [0] 1
%e A337997 [1] 0,    1
%e A337997 [2] 0,    2,      8
%e A337997 [3] 0,    6,     48,      162
%e A337997 [4] 0,   24,    384,     1944,     6144
%e A337997 [5] 0,  120,   3840,    29160,   122880,    375000
%e A337997 [6] 0,  720,  46080,   524880,  2949120,  11250000,   33592320
%e A337997 [7] 0, 5040, 645120, 11022480, 82575360, 393750000, 1410877440, 4150656720
%p A337997 # Two alternative implementations are given in the link.
%p A337997 GeneralizedEulerianPolynomial := proc(n, k, x) local S;
%p A337997    if n = 0 then  return 1 fi;
%p A337997    S := m -> add((-1)^j*binomial(n+1,j)*(k*(m-j)+1)^n*signum(k*(m-j)+1),j=0..n+1);
%p A337997    add(S(m)*x^m, m=0..n)/2 end:
%p A337997 T := (n, k) -> subs(x=1, GeneralizedEulerianPolynomial(n, k, x)):
%p A337997 for n from 0 to 6 do seq(T(n, k), k=0..n) od;
%Y A337997 Cf. A225116, A225117, A225118, A337996, A008292, A060187, A173018, A123125, A284861.
%K A337997 nonn,tabl
%O A337997 0,5
%A A337997 _Peter Luschny_, Oct 07 2020