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
Toufik Mansour, Howard Skogman, Rebecca Smith, Passing through a stack k times with reversals, arXiv:1808.04199 [math.CO], 2018.
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
KEYWORD
sign
AUTHOR
Roger L. Bagula, Aug 07 2009
STATUS
approved