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 A117443 #18 Jun 02 2021 22:17:35
%S A117443 1,0,2,-4,28,-160,1272,-11184,114448,-1309440,16680992,-233587264,
%T A117443 3569157568,-59075960320,1053056675712,-20111857791744,
%U A117443 409715696197888,-8868323731660800,203247024658514432,-4916860703228314624,125206830774036241408,-3347784042587048058880
%N A117443 Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).
%C A117443 Row sums of number triangle A117442. Binomial transform of alternating sign Springer numbers (-1)^n*A001586(n).
%H A117443 G. C. Greubel, <a href="/A117443/b117443.txt">Table of n, a(n) for n = 0..425</a>
%F A117443 E.g.f.: 1/Q(0); Q(k)=1-(x^2)/((4*k+1)*(2*k+1)+2*x*(4*k+1)*(2*k+1)/(4*k+3-2*x-x*(4*k+3)/(x-(4*k+4)/Q(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 28 2011
%F A117443 G.f.: 1/Q(0) where Q(k) = 1 + 4*k*x - 2*x^2*(2*k + 1)^2/( 1 + (4*k+2)*x - 2*x^2*(2*k + 2)^2/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 10 2013
%F A117443 a(n) ~ (-1)^n * n! * 2^(2*n+3/2) / (Pi^(n+1) * exp(Pi/4)). - _Vaclav Kotesovec_, Aug 04 2014
%F A117443 a(n) = Sum_{k=0..n} (-1)^(n+k) * binomial(n, k) * abs(numerator( Euler(n-k, 1/4) )), where Euler(n, x) is the Euler number polynomial. - _G. C. Greubel_, Jun 02 2021
%t A117443 CoefficientList[Series[E^x/(Cos[x]+Sin[x]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 04 2014 *)
%t A117443 A117442[n_, k_]:= (-1)^(n-k)*Binomial[n, k]*Abs[Numerator[EulerE[n-k, 1/4]]]; Table[Sum[A117442[n, k], {k, 0, n}], {n, 0, 30}] (* _G. C. Greubel_, Jun 02 2021 *)
%o A117443 (Sage)
%o A117443 @CachedFunction
%o A117443 def f(n): return (-1/4)^n*sum( binomial(n, j)*2^j*euler_number(j) for j in (0..n) ) # f(n) = Euler(n, 1/4)
%o A117443 def A117443(n): return sum( (-1)^(n+k)*binomial(n,k)*abs(numerator(f(n-k))) for k in (0..n) )
%o A117443 [A117443(n) for n in (0..30)] # _G. C. Greubel_, Jun 02 2021
%o A117443 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x)/(cos(x) + sin(x)))) \\ _Michel Marcus_, Jun 02 2021
%Y A117443 Cf. A117442.
%K A117443 easy,sign
%O A117443 0,3
%A A117443 _Paul Barry_, Mar 16 2006