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 A358603 #16 Jul 25 2024 14:52:58
%S A358603 1,1,0,-1,0,3,2,-9,-12,35,78,-153,-544,723,4170,-3337,-35028,10851,
%T A358603 320678,57255,-3178152,-2190253,33864546,42120183,-385314460,
%U A358603 -719159517,4649508222,12033407591,-59076411312,-204022615725,784134861818,3554417974647,-10768948801764
%N A358603 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-k)!/(n-2*k)!.
%H A358603 Seiichi Manyama, <a href="/A358603/b358603.txt">Table of n, a(n) for n = 0..899</a>
%F A358603 a(n) = (a(n-1) - n * a(n-2) + 1)/2 for n > 1.
%t A358603 nxt[{n_,a_,b_}]:={n+1,b,(b-a(n+1)+1)/2}; NestList[nxt,{1,1,1},40][[;;,2]] (* _Harvey P. Dale_, Jul 25 2024 *)
%o A358603 (PARI) a(n) = sum(k=0, n\2, (-1)^k*(n-k)!/(n-2*k)!);
%Y A358603 Cf. A358604, A358605, A358606.
%Y A358603 Cf. A122852.
%K A358603 sign
%O A358603 0,6
%A A358603 _Seiichi Manyama_, Nov 23 2022