This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373389 #10 Jun 09 2024 09:07:12
%S A373389 1,1,1,-1,-1,7,1,-103,197,2143,-11717,-50245,700541,445297,-46679363,
%T A373389 145710035,3366506123,-28627646249,-232615347479,4589590326917,
%U A373389 7797081908429,-722997025420733,2790363142173367,112843029882305495,-1235970846163474579,-16017081358111849247
%N A373389 The Eulerian zig-zag polynomials A205497 evaluated at x = -1/2 and normalized by (-2)^n.
%p A373389 # Using the recurrence by _Kyle Petersen_ from A205497.
%p A373389 G := proc(n) option remember; local F;
%p A373389 if n = 0 then 1/(1 - q*x) else F := G(n - 1);
%p A373389 simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
%p A373389 A373389 := n -> (-2)^n*subs({p = 1, q = 1, x = -1/2}, G(n)*(1 - x)^(n + 1)):
%p A373389 seq(A373389(n), n = 0..22);
%t A373389 G[n_] := G[n] = Module[{F}, If[n == 0, 1/(1-q*x), F = G[n-1]; Simplify[ (p/(p-q))*(ReplaceAll[F, {p -> q, q -> p}] - ReplaceAll[F, p -> q])]]];
%t A373389 a[n_] := a[n] = (-2)^n*ReplaceAll[G[n]*(1-x)^(n+1), {p -> 1, q -> 1, x -> -1/2}];
%t A373389 Table[Print[n, " ", a[n]]; a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jun 08 2024, after Maple program *)
%Y A373389 Cf. A000111, A205497, A350354, A373388.
%K A373389 sign
%O A373389 0,6
%A A373389 _Peter Luschny_, Jun 03 2024