login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Robert Israel, Table of n, a(n) for n = 0..485

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(2EulerE(n,1)+EulerE(n,3/2)))/2 +1, where EulerE(n,x) is the nth Euler polynomial. - Benedict W. J. Irwin, May 24 2016

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

Cf. A000111.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 8 04:27 EST 2016. Contains 278902 sequences.