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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.

%I A373431 #9 Jun 06 2024 04:23:44
%S A373431 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,
%T A373431 2476,2476,1229,291,29,1,1,47,760,5303,18415,33818,33818,18415,5303,
%U A373431 760,47,1,1,74,1818,19481,106272,317902,544727,544727,317902,106272,19481,1818,74,1
%N 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.
%C A373431 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.
%H A373431 Peter Luschny, <a href="/A373431/a373431.png">Illustrating the polynomials</a>.
%e A373431 Triangle starts:
%e A373431   [0] 1;
%e A373431   [1] 1;
%e A373431   [2] 1,  1;
%e A373431   [3] 1,  4,   4,    1;
%e A373431   [4] 1,  9,  25,   25,    9,    1;
%e A373431   [5] 1, 17,  97,  221,  221,   97,   17,   1;
%e A373431   [6] 1, 29, 291, 1229, 2476, 2476, 1229, 291, 29, 1;
%p A373431 R := proc(n) option remember; local F; if n = 0 then 1/(1 - q*x) else F := R(n-1);
%p A373431 simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
%p A373431 EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1 - x)^(n + 1)) / x^2)):
%p A373431 nc := (n, k) -> `if`(n = 0, 0^n, binomial(n, k)^2*(n-k)/(n*(k+1))):
%p A373431 N := (n, x) -> local k; simplify(add(nc(n, k)*x^k, k = 0..n)):
%p A373431 NEZ := (n, x) -> expand(EZ(n, x) * N(n, x)):
%p A373431 Trow := n -> local k; if n < 2 then 1 elif n = 2 then 1, 1
%p A373431 else seq(coeff(NEZ(n, x), x, k), k = 0..2*n-3) fi: seq(print(Trow(n)), n = 0..6);
%Y A373431 Cf. A131198 (Narayana), A205497 (Eulerian zig-zag), A373430 (row sums).
%K A373431 nonn,tabf
%O A373431 0,6
%A A373431 _Peter Luschny_, Jun 05 2024