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 A163747 E.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)). 9
 0, -1, -1, 2, 5, -16, -61, 272, 1385, -7936, -50521, 353792, 2702765, -22368256, -199360981, 1903757312, 19391512145, -209865342976, -2404879675441, 29088885112832, 370371188237525, -4951498053124096, -69348874393137901 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. From Paul Curtz, Mar 12 2013: (Start) a(n) is an autosequence of the first kind; a(n) and successive differences are: 0,   -1,   -1,      2,      5,    -16,    -61; -1,   0,    3,      3,    -21,    -45,    333; 1,    3,    0,    -24,    -24,    378,    780; 2,   -3,  -24,      0,    402,    402, -11214; -5, -21,   24,    402,      0, -11616, -11616; -16, 45,  378,   -402, -11616,      0, 514608; 61, 333, -780, -11214,  11616, 514608,      0; The main diagonal is A000004. The inverse binomial transform is the signed sequence. The first two upper diagonals are A002832 (median Euler numbers) signed. Sum of the antidiagonals: 0, -2, 0, 10, 0, ... = 2*A122045(n+1) (End) LINKS Robert Israel, Table of n, a(n) for n = 0..485 Toufik Mansour, Howard Skogman, and Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018. A. Randrianarivony and J. Zeng, Une famille de polynomes qui interpole plusieurs suites classiques de nombres, Adv. Appl. Math. 17 (1996), 1-26. See Section 6 (negative of the zeroth column of matrix a_{n,k} on p. 18). FORMULA 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 a(n) ~ n! * (cos(Pi*n/2) - sin(Pi*n/2)) * 2^(n+2) / Pi^(n+1). - Vaclav Kotesovec, Apr 23 2015 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 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 a(n) = 2^n*(Euler(n, 1/2) - Euler(n, 1)). - Peter Luschny, Mar 19 2021 a(n) = ((-2)^(n + 1)*(1 - 2^(n + 1))*Bernoulli(n + 1))/(n + 1) + Euler(n). - Peter Luschny, May 06 2021 MAPLE 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 seq((euler(n) - 2^n*(2*euler(n, 1)+euler(n, 3/2)))/2 + 1, n=0..30); # Robert Israel, May 24 2016 MATHEMATICA f[t_] = (1 + I)/(1 + I*Exp[t]) - 1 Table[Re[2*n!*SeriesCoefficient[ Series[f[t], {t, 0, 30}], n]], {n, 0, 30}] 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 *) 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 *) CROSSREFS Variant: A163982. Cf. A000004, A000111, A002832, A122045. Minus the zeroth column of A323833. Sequence in context: A178123 A138265 A275711 * A000111 A007976 A058259 Adjacent sequences:  A163744 A163745 A163746 * A163748 A163749 A163750 KEYWORD sign AUTHOR Roger L. Bagula, Aug 03 2009 STATUS approved

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Last modified June 14 08:31 EDT 2021. Contains 345018 sequences. (Running on oeis4.)