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 A373388 #16 Jun 09 2024 13:20:08
%S A373388 1,-1,1,0,-1,0,5,0,-45,0,665,0,-14457,0,433741,0,-17160421,0,
%T A373388 865407905,0,-54179057649,0,4122477869077,0,-374673778941981,0,
%U A373388 40087507726395689,0,-4987405802167886825,0,713925031978621041757,0,-116506260029721326349781,0,21501227314690679723073329
%N A373388 Alternating row sums of the Eulerian zig-zag number triangle A205497.
%p A373388 # Using the recurrence by _Kyle Petersen_ from A205497.
%p A373388 G := proc(n) option remember; local F;
%p A373388 if n = 0 then 1/(1 - q*x) else F := G(n - 1);
%p A373388 simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
%p A373388 A373388 := n -> subs({p = 1, q = 1, x = -1}, G(n)*(1 - x)^(n + 1)):
%p A373388 seq(A373388(n), n = 0..20);
%t A373388 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])]]]; a[n_] := a[n] = ReplaceAll[G[n]*(1-x)^(n+1), {p -> 1, q -> 1, x -> -1}]; Table[a[n], {n, 0, 34}] (* _Jean-François Alcover_, Jun 08 2024, after Maple program *)
%Y A373388 Cf. A000111, A205497, A373389.
%K A373388 sign
%O A373388 0,7
%A A373388 _Peter Luschny_, Jun 03 2024