A373434 Triangle read by rows: Coefficients of the Eulerian polynomials EC(n, x)*EZ(n, x), where EC denote the classical Eulerian and EZ the zig-zag Eulerian polynomials.

1, 1, 1, 1, 1, 5, 5, 1, 1, 14, 45, 45, 14, 1, 1, 33, 255, 671, 671, 255, 33, 1, 1, 71, 1131, 6311, 14446, 14446, 6311, 1131, 71, 1, 1, 146, 4420, 46571, 206932, 427370, 427370, 206932, 46571, 4420, 146, 1, 1

Offset

0,6

Comments

There are various conventions for indexing Eulerian numbers. The one used here determines that all corresponding polynomials have p(n, 0) = 1. This applies equally to the classical Eulerian polynomials with coefficients A173018, the Eulerian zig-zag polynomials with coefficients A205497, and the polynomials here.

Links

Table of n, a(n) for n=0..44.

Peter Luschny, Illustrating the polynomials.

Example

Triangle T(n, k) starts:
  [0] 1;
  [1] 1;
  [2] 1,  1;
  [3] 1,  5,    5,    1;
  [4] 1, 14,   45,   45,    14,     1;
  [5] 1, 33,  255,  671,   671,   255,   33,    1;
  [6] 1, 71, 1131, 6311, 14446, 14446, 6311, 1131, 71,  1;
  ...
Written as polynomials P(n, x):
  [0] 1;
  [1] 1;
  [2] 1 +    x;
  [3] 1 +  5*x +   5*x^2 +     x^3;
  [4] 1 + 14*x +  45*x^2 +  45*x^3 +  14*x^4 +     x^5;
  [5] 1 + 33*x + 255*x^2 + 671*x^3 + 671*x^4 + 255*x^5 + 33*x^6 + x^7;
  ...
P(3, x) = A205497(3, x) * A173018(3, x) = (1 + x) * (1 + 4*x + x^2) = 1 + 5*x + 5*x^2 + x^3.

Maple

# Using the recurrence by Kyle Petersen from A205497.
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)):
EC := (n, x) -> local k; simplify(add(combinat:-eulerian1(n, k)*x^k, k = 0..n)):
EZC := (n, x) -> expand(EZ(n, x) * EC(n, x)):
Trow := n -> local k; if n < 2 then [1] elif n = 2 then [1, 1] else [seq(coeff(EZC(n, x), x, k), k = 0..2*n-3)] fi:
seq(print(EZC(n, x)), n = 0..6); seq(print(Trow(n)), n = 0..6);

Mathematica

R[n_] := R[n] = Module[{F}, If[n == 0, 1/(1 - q*x), F = R[n - 1]; Simplify[p/(p - q)*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]];
EZ[n_, x_] := If[n < 3, 1, Expand[Simplify[ReplaceAll[R[n], {p -> 1, q -> 1}]*(1 - x)^(n + 1)] / x^2]];
eulerian1[n_, k_] := If[n == 0, 1, Sum[(-1)^j*Binomial[n + 1, j]*(k + 1 - j)^n, {j, 0, k + 1}]];
EC[n_, x_] :=  Sum[eulerian1[n, k]*x^k, {k, 0, n}];
EZC [n_, x_] := Expand[EZ[n, x] * EC[n, x]];
Trow[n_] := CoefficientList[EZC[n, x], x];
Table[Trow[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 07 2024, after Peter Luschny's Maple program *)

Crossrefs

Cf. A173018 (Eulerian), A205497 (Eulerian zig-zag), A373433 (row sums).

Sequence in context: A172349 A144403 A188587 * A174119 A156696 A232651

Adjacent sequences: A373431 A373432 A373433 * A373435 A373436 A373437

Keyword

nonn,tabf

Author

Peter Luschny, Jun 04 2024

Status

approved