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%I #22 Jun 13 2024 04:44:56
%S 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,
%T 20,130,430,845,1052,845,430,130,20,1,1,33,312,1453,4023,7218,8736,
%U 7218,4023,1453,312,33,1,1,54,730,4639,17316,42142,70593,83610,70593,42142,17316,4639,730,54,1
%N 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.
%C 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).
%H Peter Luschny, <a href="/A373432/a373432.png">Illustrating the polynomials</a>.
%e Triangle starts:
%e [0] [1]
%e [1] [1, 1]
%e [2] [1, 2, 1]
%e [3] [1, 4, 6, 4, 1]
%e [4] [1, 7, 19, 26, 19, 7, 1]
%e [5] [1, 12, 52, 116, 150, 116, 52, 12, 1]
%e [6] [1, 20, 130, 430, 845, 1052, 845, 430, 130, 20, 1]
%p EZP := proc(P, len) local R, EZ, EP, EZP, CL, n;
%p R := proc(n) option remember; local F; if n = 0 then 1/(1-q*x) else F := R(n-1);
%p simplify(p/(p - q)*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
%p EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1-x)^(n+1))/x^2)):
%p EP := (n, x) -> local k; simplify(add(P(n, k)*x^k, k = 0..n)):
%p EZP := (n, x) -> expand(EZ(n, x) * EP(n, x)):
%p CL := p -> PolynomialTools:-CoefficientList(p, x);
%p seq(CL(EZP(n, x)), n = 0..len); ListTools:-Flatten([%]) end:
%p EZP(binomial, 8);
%Y Cf. A000831 (row sums), A007318 (Pascal), A205497 (zig-zag Eulerian).
%K nonn,tabf
%O 0,5
%A _Peter Luschny_, Jun 05 2024
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