A373431 Triangle read by rows: Coefficients of the polynomials N(n, x) * EZ(n, x), where N denote the Narayana polynomials A131198 and EZ the Eulerian zig-zag polynomials A205497.

1, 1, 1, 1, 1, 4, 4, 1, 1, 9, 25, 25, 9, 1, 1, 17, 97, 221, 221, 97, 17, 1, 1, 29, 291, 1229, 2476, 2476, 1229, 291, 29, 1, 1, 47, 760, 5303, 18415, 33818, 33818, 18415, 5303, 760, 47, 1, 1, 74, 1818, 19481, 106272, 317902, 544727, 544727, 317902, 106272, 19481, 1818, 74, 1

Offset: 0

Author

Peter Luschny, Jun 05 2024

Keywords

• nonn
• tabf

Comments

There are various conventions for indexing the Narayana, the Eulerian numbers and the zig-zag Eulerian numbers. The one we use here requires that all corresponding polynomials have p(n, 0) = 1.

Examples

			Triangle starts:
  [0] 1;
  [1] 1;
  [2] 1,  1;
  [3] 1,  4,   4,    1;
  [4] 1,  9,  25,   25,    9,    1;
  [5] 1, 17,  97,  221,  221,   97,   17,   1;
  [6] 1, 29, 291, 1229, 2476, 2476, 1229, 291, 29, 1;
		

Links

• Peter Luschny, Illustrating the polynomials.

Crossrefs

Cf. A131198 (Narayana), A205497 (Eulerian zig-zag), A373430 (row sums).

Programs

Maple

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)):
nc := (n, k) -> `if`(n = 0, 0^n, binomial(n, k)^2*(n-k)/(n*(k+1))):
N := (n, x) -> local k; simplify(add(nc(n, k)*x^k, k = 0..n)):
NEZ := (n, x) -> expand(EZ(n, x) * N(n, x)):
Trow := n -> local k; if n < 2 then 1 elif n = 2 then 1, 1
else seq(coeff(NEZ(n, x), x, k), k = 0..2*n-3) fi: seq(print(Trow(n)), n = 0..6);