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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.

%I A373434 #16 Jun 07 2024 08:03:43
%S A373434 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,
%T A373434 6311,14446,14446,6311,1131,71,1,1,146,4420,46571,206932,427370,
%U A373434 427370,206932,46571,4420,146,1,1
%N 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.
%C A373434 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.
%H A373434 Peter Luschny, <a href="/A373434/a373434.png">Illustrating the polynomials</a>.
%e A373434 Triangle T(n, k) starts:
%e A373434   [0] 1;
%e A373434   [1] 1;
%e A373434   [2] 1,  1;
%e A373434   [3] 1,  5,    5,    1;
%e A373434   [4] 1, 14,   45,   45,    14,     1;
%e A373434   [5] 1, 33,  255,  671,   671,   255,   33,    1;
%e A373434   [6] 1, 71, 1131, 6311, 14446, 14446, 6311, 1131, 71,  1;
%e A373434   ...
%e A373434 Written as polynomials P(n, x):
%e A373434   [0] 1;
%e A373434   [1] 1;
%e A373434   [2] 1 +    x;
%e A373434   [3] 1 +  5*x +   5*x^2 +     x^3;
%e A373434   [4] 1 + 14*x +  45*x^2 +  45*x^3 +  14*x^4 +     x^5;
%e A373434   [5] 1 + 33*x + 255*x^2 + 671*x^3 + 671*x^4 + 255*x^5 + 33*x^6 + x^7;
%e A373434   ...
%e A373434 P(3, x) = A205497(3, x) * A173018(3, x) = (1 + x) * (1 + 4*x + x^2) = 1 + 5*x + 5*x^2 + x^3.
%p A373434 # Using the recurrence by _Kyle Petersen_ from A205497.
%p A373434 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:
%p A373434 EZ := (n, x) -> ifelse(n < 3, 1, expand(simplify(subs({p = 1, q = 1}, R(n))*(1 - x)^(n + 1)) / x^2)):
%p A373434 EC := (n, x) -> local k; simplify(add(combinat:-eulerian1(n, k)*x^k, k = 0..n)):
%p A373434 EZC := (n, x) -> expand(EZ(n, x) * EC(n, x)):
%p A373434 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:
%p A373434 seq(print(EZC(n, x)), n = 0..6); seq(print(Trow(n)), n = 0..6);
%t A373434 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])]]];
%t A373434 EZ[n_, x_] := If[n < 3, 1, Expand[Simplify[ReplaceAll[R[n], {p -> 1, q -> 1}]*(1 - x)^(n + 1)] / x^2]];
%t A373434 eulerian1[n_, k_] := If[n == 0, 1, Sum[(-1)^j*Binomial[n + 1, j]*(k + 1 - j)^n, {j, 0, k + 1}]];
%t A373434 EC[n_, x_] :=  Sum[eulerian1[n, k]*x^k, {k, 0, n}];
%t A373434 EZC [n_, x_] := Expand[EZ[n, x] * EC[n, x]];
%t A373434 Trow[n_] := CoefficientList[EZC[n, x], x];
%t A373434 Table[Trow[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jun 07 2024, after _Peter Luschny_'s Maple program *)
%Y A373434 Cf. A173018 (Eulerian), A205497 (Eulerian zig-zag), A373433 (row sums).
%K A373434 nonn,tabf
%O A373434 0,6
%A A373434 _Peter Luschny_, Jun 04 2024