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A163747 Expansion of e.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)).

%I A163747 #67 Aug 09 2021 15:43:56
%S A163747 0,-1,-1,2,5,-16,-61,272,1385,-7936,-50521,353792,2702765,-22368256,
%T A163747 -199360981,1903757312,19391512145,-209865342976,-2404879675441,
%U A163747 29088885112832,370371188237525,-4951498053124096,-69348874393137901
%N A163747 Expansion of e.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)).
%C A163747 The real part of the exponential expansion of 2*((1+i)/(1+i*exp(z))-1) = (-1-i)*z + (-1/2+i/2)*z^2 + (1/3+i/3)*z^3 + (5/24-5i/24)*z^4 + (-2/15-2i/15)*z^5 + ...  where i is the imaginary unit.
%C A163747 From _Paul Curtz_, Mar 12 2013: (Start)
%C A163747 a(n) is an autosequence of the first kind; a(n) and successive differences are:
%C A163747 0,   -1,   -1,      2,      5,    -16,    -61;
%C A163747 -1,   0,    3,      3,    -21,    -45,    333;
%C A163747 1,    3,    0,    -24,    -24,    378,    780;
%C A163747 2,   -3,  -24,      0,    402,    402, -11214;
%C A163747 -5, -21,   24,    402,      0, -11616, -11616;
%C A163747 -16, 45,  378,   -402, -11616,      0, 514608;
%C A163747 61, 333, -780, -11214,  11616, 514608,      0;
%C A163747 The main diagonal is A000004. The inverse binomial transform is the signed sequence.
%C A163747 The first two upper diagonals are A002832 (median Euler numbers) signed.
%C A163747 Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End)
%H A163747 Robert Israel, <a href="/A163747/b163747.txt">Table of n, a(n) for n = 0..485</a>
%H A163747 Toufik Mansour, Howard Skogman, and 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.
%H A163747 A. Randrianarivony and J. Zeng, <a href="http://dx.doi.org/10.1006/aama.1996.0001">Une famille de polynomes qui interpole plusieurs suites classiques de nombres</a>, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (negative of the zeroth column of matrix a_{n,k} on p. 18).
%F A163747 G.f.: -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 A163747 a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Apr 23 2015
%F A163747 a(n) = (A122045(n) - 2^n(2*Euler(n,1) + Euler(n,3/2)))/2 + 1, where Euler(n,x) is the n-th Euler polynomial. - _Benedict W. J. Irwin_, May 24 2016
%F A163747 a(n) = 2*4^n*(HurwitzZeta(-n, 1/4) - HurwitzZeta(-n, 3/4)) + HurwitzZeta(-n, 1)*(4^(n+1) - 2^(n+1)). - _Peter Luschny_, Jul 21 2020
%F A163747 a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - _Peter Luschny_, Mar 19 2021
%F A163747 a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - _Peter Luschny_, May 06 2021
%F A163747 a(n) = n!*Re([x^n]((2 - 2*i)/(i + exp(-x)))). - _Peter Luschny_, Aug 09 2021
%p A163747 A163747 := proc(n) exp(t)*(1-exp(t))/(1+exp(2*t)) ; coeftayl(%,t=0,n) ; 2*%*n! ; end proc: # _R. J. Mathar_, Sep 11 2011
%p A163747 seq((euler(n) - 2^n*(2*euler(n,1)+euler(n,3/2)))/2 + 1, n=0..30); # _Robert Israel_, May 24 2016
%p A163747 egf := (2 - 2*I)/(exp(-x) + I); ser := series(egf, x, 24):
%p A163747 seq(n!*Re(coeff(ser, x, n)), n = 0..22); # _Peter Luschny_, Aug 09 2021
%t A163747 f[t_] = (1 + I)/(1 + I*Exp[t]) - 1;
%t A163747 Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]
%t A163747 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 = -x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* _Vaclav Kotesovec_, Jan 22 2015, after _Sergei N. Gladkovskii_ *)
%t A163747 Table[(EulerE[n] - 2^n (2 EulerE[n, 1] + EulerE[n, 3/2]))/2 + 1, {n, 0, 20}] (* _Benedict W. J. Irwin_, May 24 2016 *)
%Y A163747 Variant: A163982.
%Y A163747 Cf. A000004, A000111, A002832, A122045.
%Y A163747 Minus the zeroth column of A323833.
%K A163747 sign
%O A163747 0,4
%A A163747 _Roger L. Bagula_, Aug 03 2009