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A163747 E.g.f. 2*exp(x)*(1-exp(x))/(1+exp(2*x)). 5
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

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

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

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}]

CROSSREFS

Cf. A000111.

Sequence in context: A104858 A178123 A138265 * 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 November 29 03:09 EST 2014. Contains 250479 sequences.