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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

Table of n, a(n) for n=0..22.

FORMULA

Expansion[(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

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 *)

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 December 4 05:11 EST 2016. Contains 278748 sequences.