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

0,6

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.

Links

Table of n, a(n) for n=0..57.

Peter Luschny, Illustrating the polynomials.

Example

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;

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);

Crossrefs

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

Sequence in context: A172347 A375863 A108428 * A174126 A075613 A221837

Adjacent sequences: A373428 A373429 A373430 * A373432 A373433 A373434

Keyword

nonn,tabf

Author

Peter Luschny, Jun 05 2024

Status

approved