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 A163982 #34 Nov 07 2018 14:44:15
%S A163982 -2,-1,1,2,-5,-16,61,272,-1385,-7936,50521,353792,-2702765,-22368256,
%T A163982 199360981,1903757312,-19391512145,-209865342976,2404879675441,
%U A163982 29088885112832,-370371188237525,-4951498053124096,69348874393137901
%N A163982 Real part of the coefficient [x^n] of the expansion of (1+i)/(1-i*exp(x)) - 1 multiplied by 2*n!, where i is the imaginary unit.
%C A163982 The sequence is a signed variant of A163747 and starts with a two instead of a zero.
%C A163982 From _Paul Curtz_, Mar 20 2013: (Start)
%C A163982 -a(n) and successive differences are:
%C A163982 2, 1, -1, -2, 5, 16, -61, -272;
%C A163982 -1, -2, -1, 7, 11, -77, -211, 1657, ...
%C A163982 -1, 1, 8, 4, -88, -134, 1868, 4894, ...
%C A163982 2, 7, -4, -92, -46, -46, 2002, 3026, ...
%C A163982 5, -11, -88, 46, 2048, 1024, -72928, ...
%C A163982 -16, -77, 134, 2002, -1024, -73952, -36976, ...
%C A163982 -61, 211, 1868, -3026, -72928, ...
%C A163982 272, 1657, -4894, -69902, ...
%C A163982 This is an autosequence: The inverse binomial transform (left column of the array of differences) is the signed sequence. The main diagonal 2, -2, 8, -92, ... doubles the entries of the first upper diagonal 1, -1, 4, -46, ... = A099023(n).
%C A163982 Sum of the antidiagonals: 2, 0, -4, 0, 32, ... = 2*A155585(n+1). (End)
%H A163982 G. C. Greubel, <a href="/A163982/b163982.txt">Table of n, a(n) for n = 0..480</a>
%H A163982 Toufik Mansour, Howard Skogman, Rebecca Smith, <a href="https://arxiv.org/abs/1808.04199">Passing through a stack k times with reversals</a>, arXiv:1808.04199 [math.CO], 2018.
%F A163982 Let ((1 + i)/(1 - i*exp(t)) - 1) = a(n) + I*b(n); abs(a(n)) = abs(b(n)).
%F A163982 a(n) = -2^n*(E_{n}(1/2) + E_{n}(1)), E_{n}(x) Euler polynomial. - _Peter Luschny_, Nov 25 2010
%F A163982 E.g.f.: -(1/cosh(x) + tanh(x)) - 1. - _Sergei N. Gladkovskii_, Dec 11 2013
%F A163982 G.f.: -2 - x/W(0), where W(k) = 1 + x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 /W(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Jan 22 2015
%F A163982 E.g.f.: (-2)*exp(x/2)*cosh(x/2)/cosh(x). - _G. C. Greubel_, Aug 24 2017
%p A163982 A163982 := n -> -2^n*(euler(n,1/2)+euler(n,1)): # _Peter Luschny_, Nov 25 2010
%p A163982 A163982 := proc(n)
%p A163982 (1+I)/(1-I*exp(x))-1 ;
%p A163982 coeftayl(%,x=0,n) ;
%p A163982 Re(%*2*n!) ;
%p A163982 end proc; # _R. J. Mathar_, Mar 26 2013
%t A163982 f[t_] = (1 + I)/(1 - I*Exp[t]) - 1; Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
%t A163982 max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 + x + (4*k+3)*(k+1)*x^2 /( 1 + (4*k+5)*(k+1)*x^2 / g[k+1]); gf = -2 - x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* _Vaclav Kotesovec_, Jan 22 2015, after _Sergei N. Gladkovskii_ *)
%t A163982 With[{nn = 50}, CoefficientList[Series[(-2)*Exp[t/2]*Cosh[t/2]/Cosh[t], {t, 0, nn}], t]*Range[0, nn]!] (* _G. C. Greubel_, Aug 24 2017 *)
%o A163982 (PARI) t='t+O('t^10); Vec(serlaplace((-2)*exp(x/2)*cosh(x/2)/cosh(x))) \\ _G. C. Greubel_, Aug 24 2017
%Y A163982 Cf. A163747.
%K A163982 sign
%O A163982 0,1
%A A163982 _Roger L. Bagula_, Aug 07 2009