OFFSET
0,5
COMMENTS
There are various conventions for indexing Eulerian numbers. The one used here is described by the condition that for all polynomials p(n, 0) = 1. This applies equally to the classical Eulerian polynomials given by the coefficients A173018, as well as to the Eulerian zig-zag polynomials with coefficients in A205497 and to the polynomials here. See the illustration (link section).
LINKS
Peter Luschny, Illustrating the polynomials.
EXAMPLE
Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [1, 4, 6, 4, 1]
[4] [1, 7, 19, 26, 19, 7, 1]
[5] [1, 12, 52, 116, 150, 116, 52, 12, 1]
[6] [1, 20, 130, 430, 845, 1052, 845, 430, 130, 20, 1]
MAPLE
EZP := proc(P, len) local R, EZ, EP, EZP, CL, n;
R := proc(n) option remember; local F; if n = 0 then 1/(1-q*x) else F := R(n-1);
simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1-x)^(n+1))/x^2)):
EP := (n, x) -> local k; simplify(add(P(n, k)*x^k, k = 0..n)):
EZP := (n, x) -> expand(EZ(n, x) * EP(n, x)):
CL := p -> PolynomialTools:-CoefficientList(p, x);
seq(CL(EZP(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
EZP(binomial, 8);
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jun 05 2024
STATUS
approved