A373432 Triangle read by rows. Coefficients of the polynomials P(n, x) * EZ(n, x), where P denote the Pascal polynomials and EZ the zig-zag Eulerian polynomials A205497.

1, 1, 1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 7, 19, 26, 19, 7, 1, 1, 12, 52, 116, 150, 116, 52, 12, 1, 1, 20, 130, 430, 845, 1052, 845, 430, 130, 20, 1, 1, 33, 312, 1453, 4023, 7218, 8736, 7218, 4023, 1453, 312, 33, 1, 1, 54, 730, 4639, 17316, 42142, 70593, 83610, 70593, 42142, 17316, 4639, 730, 54, 1

Offset: 0

Author

Peter Luschny, Jun 05 2024

Keywords

• nonn
• tabf

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).

Examples

			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]
		

Links

• Peter Luschny, Illustrating the polynomials.

Crossrefs

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

Programs

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);