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 A165233 #20 Mar 01 2024 02:03:09
%S A165233 1,1,-2,-6,24,120,-720,-5040,40320,362880,-3628800,-39916800,
%T A165233 479001600,6227020800,-87178291200,-1307674368000,20922789888000,
%U A165233 355687428096000,-6402373705728000,-121645100408832000,2432902008176640000
%N A165233 Signed denominators of terms in series expansion of cos(x)+sin(x).
%C A165233 Sum(n>=0,1/a(n))=cos(1)+sin(1).
%C A165233 Sum(n>=0,(Pi/4)^n/a(n))=sqrt(2).
%C A165233 Numerators are in A000012. - _Alois P. Heinz_, Jan 20 2016
%F A165233 a(n)=(-1)^floor(n/2)*n! = A057077(n)*A000142(n).
%F A165233 a(n)=(sin(n*Pi/2)+cos(n*Pi/2))*n!.
%F A165233 a(n)=sqrt(2)*sin((2n+1)*Pi/4)*n!.
%F A165233 a(n)=sqrt(2)*cos((2n-1)*Pi/4)*n!.
%F A165233 G.f. Q(0) where Q(k)= 1 + x*(4*k+1)/(1 + 2*x*(2*k+1)/(1 - 2*x*(2*k+1) - x*(4*k+3)/(1 + x*(4*k+3) - 4*x*(k+1)/(4*x*(k+1) - 1/Q(k+1))))); (continued fraction, 3rd kind, 5-step). - _Sergei N. Gladkovskii_, Aug 15 2012
%F A165233 E.g.f.: (1 + x)/(1 + x^2). - _Ilya Gutkovskiy_, Oct 08 2016
%t A165233 Sign@ # Denominator@ # & /@ CoefficientList[Series[Cos@ x + Sin@ x, {x, 0, 20}], x] (* _Michael De Vlieger_, Oct 08 2016 *)
%o A165233 (PARI) a(n)=(-1)^(n\2)*n!
%Y A165233 Cf. A000012, A000142, A133942, A057077, A143623, A002193.
%K A165233 frac,sign,easy
%O A165233 0,3
%A A165233 _Jaume Oliver Lafont_, Sep 09 2009