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

0,5

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

Table of n, a(n) for n=0..65.

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

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

Keyword

nonn,tabf

Author

Peter Luschny, Jun 05 2024

Status

approved