%I #67 Aug 09 2021 15:43:56

%S 0,-1,-1,2,5,-16,-61,272,1385,-7936,-50521,353792,2702765,-22368256,

%T -199360981,1903757312,19391512145,-209865342976,-2404879675441,

%U 29088885112832,370371188237525,-4951498053124096,-69348874393137901

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

%C 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 From _Paul Curtz_, Mar 12 2013: (Start)

%C a(n) is an autosequence of the first kind; a(n) and successive differences are:

%C 0, -1, -1, 2, 5, -16, -61;

%C -1, 0, 3, 3, -21, -45, 333;

%C 1, 3, 0, -24, -24, 378, 780;

%C 2, -3, -24, 0, 402, 402, -11214;

%C -5, -21, 24, 402, 0, -11616, -11616;

%C -16, 45, 378, -402, -11616, 0, 514608;

%C 61, 333, -780, -11214, 11616, 514608, 0;

%C The main diagonal is A000004. The inverse binomial transform is the signed sequence.

%C The first two upper diagonals are A002832 (median Euler numbers) signed.

%C Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End)

%H Robert Israel, <a href="/A163747/b163747.txt">Table of n, a(n) for n = 0..485</a>

%H 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 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 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 a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - _Vaclav Kotesovec_, Apr 23 2015

%F 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 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 a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - _Peter Luschny_, Mar 19 2021

%F a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - _Peter Luschny_, May 06 2021

%F a(n) = n!*Re([x^n]((2 - 2*i)/(i + exp(-x)))). - _Peter Luschny_, Aug 09 2021

%p 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 seq((euler(n) - 2^n*(2*euler(n,1)+euler(n,3/2)))/2 + 1, n=0..30); # _Robert Israel_, May 24 2016

%p egf := (2 - 2*I)/(exp(-x) + I); ser := series(egf, x, 24):

%p seq(n!*Re(coeff(ser, x, n)), n = 0..22); # _Peter Luschny_, Aug 09 2021

%t f[t_] = (1 + I)/(1 + I*Exp[t]) - 1;

%t Table[Re[2*n!*SeriesCoefficient[Series[f[t], {t, 0, 30}], n]], {n, 0, 30}]

%t 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 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 Variant: A163982.

%Y Cf. A000004, A000111, A002832, A122045.

%Y Minus the zeroth column of A323833.

%K sign

%O 0,4

%A _Roger L. Bagula_, Aug 03 2009