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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. 3
-2, -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

0,1

COMMENTS

The sequence is a signed variant of A163747 and starts with a two instead of a zero.

From Paul Curtz, Mar 20 2013: (Start)

-a(n) and successive differences are:

    2,    1,    -1,     -2,      5,     16,    -61, -272;

   -1,   -2,    -1,      7,     11,    -77,   -211, 1657, ...

   -1,    1,     8,      4,    -88,   -134,   1868, 4894, ...

    2,    7,    -4,    -92,    -46,    -46,   2002, 3026, ...

    5,  -11,   -88,     46,   2048,   1024, -72928, ...

  -16,  -77,   134,   2002,  -1024, -73952, -36976, ...

  -61,  211,  1868,  -3026, -72928, ...

  272, 1657, -4894, -69902, ...

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).

Sum of the antidiagonals: 2, 0, -4, 0, 32, ... = 2*A155585(n+1). (End)

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..480

FORMULA

Let ((1 + i)/(1 - i*exp(t)) - 1) = a(n) + I*b(n); abs(a(n)) = abs(b(n)).

a(n) = -2^n*(E_{n}(1/2) + E_{n}(1)), E_{n}(x) Euler polynomial. - Peter Luschny, Nov 25 2010

E.g.f.: -(1/cosh(x) + tanh(x)) - 1. - Sergei N. Gladkovskii, Dec 11 2013

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

E.g.f.: (-2)*exp(x/2)*cosh(x/2)/cosh(x). - G. C. Greubel, Aug 24 2017

MAPLE

A163982 := n -> -2^n*(euler(n, 1/2)+euler(n, 1)): # Peter Luschny, Nov 25 2010

A163982 := proc(n)

    (1+I)/(1-I*exp(x))-1 ;

    coeftayl(%, x=0, n) ;

    Re(%*2*n!) ;

end proc; # R. J. Mathar, Mar 26 2013

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 = -2 - x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jan 22 2015, after Sergei N. Gladkovskii *)

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 *)

PROG

(PARI) t='t+O('t^10); Vec(serlaplace((-2)*exp(x/2)*cosh(x/2)/cosh(x))) \\ G. C. Greubel, Aug 24 2017

CROSSREFS

Cf. A163747.

Sequence in context: A273488 A117848 A025242 * A246661 A246660 A245405

Adjacent sequences:  A163979 A163980 A163981 * A163983 A163984 A163985

KEYWORD

sign

AUTHOR

Roger L. Bagula, Aug 07 2009

STATUS

approved

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Last modified October 24 03:55 EDT 2017. Contains 293834 sequences.