A098276 - OEIS

%I #18 Sep 29 2013 21:16:48

%S 1,0,2,0,72,-320,3600,-32256,344960,-3926016,48625920,-648243200,

%T 9270125568,-141579509760,2300668418048,-39642283376640,

%U 722055883161600,-13863472939925504,279868860012625920

%N Difference between the number of even reduced Latin rectangles of size 3 X n and the number of odd ones.

%H Vincenzo Librandi, <a href="/A098276/b098276.txt">Table of n, a(n) for n = 1..200</a>

%H J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">The generating function for the difference in even and odd three-line latin rectangles</a>, Ann. Sci. Math. Quebec 20/1 (1996) 105-108. [<a href="http://www.labmath.uqam.ca/~annales/volumes/20-1/PDF/105-108.pdf">alternate link</a>]

%F E.g.f.: x + [(1-x)^2/(1+x)+x/(1+x)^2] * exp(2x). - corrected by _Vaclav Kotesovec_, Sep 29 2013

%F a(n) ~ n! * (-1)^(n+1) * n * exp(-2). - _Vaclav Kotesovec_, Sep 29 2013

%t Rest[CoefficientList[Series[x + ((1-x)^2/(1+x)+x/(1+x)^2)*E^(2*x), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Sep 29 2013 *)

%o (PARI) a(n)=polcoeff(serlaplace(exp(2*x)*((1-x)^2/(1+x)+x/(1+x)^2)),n)

%o (PARI) a(n)=(-1)^(n-1)*(n-2)*n!/2*polcoeff(Ser(exp(2*(atanh(x)-x))),n)

%Y Cf. A000186.

%K sign,easy

%O 1,3

%A _Ralf Stephan_, Sep 06 2004