0,5

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).

Peter Luschny, Illustrating the polynomials.

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]

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);

Cf. A000831 (row sums), A007318 (Pascal), A205497 (zig-zag Eulerian).

Sequence in context: A273891 A034870 A324224 * A264622 A275017 A141036

Adjacent sequences: A373429 A373430 A373431 * A373433 A373434 A373435

nonn,tabf

Peter Luschny, Jun 05 2024

approved