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%I #41 Dec 20 2023 15:57:29
%S 1,0,-16,-64,-160,-320,-560,-896,-1344,-1920,-2640,-3520,-4576,-5824,
%T -7280,-8960,-10880,-13056,-15504,-18240,-21280,-24640,-28336,-32384,
%U -36800,-41600,-46800,-52416,-58464,-64960,-71920,-79360,-87296,-95744,-104720,-114240,-124320,-134976,-146224,-158080
%N Expansion of g.f. (1+x)^2*(x^2-6*x+1)/(x-1)^4.
%C This g.f. is the eighth power of the spontaneous magnetization series for the two-dimensional square lattice in the parameter x = exp(-4J/kT), cf. A002928.
%D Terrel L. Hill, Statistical Mechanics: Principles and Selected Applications, Dover, New York, 1956, page 331. See eq. 44.12 for the g.f. with x replaced by x^2.
%H M. R. Sepanski, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i1p32">On Divisibility of Convolutions of Central Binomial Coefficients</a>, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 8*n*(1 - n^2)/3, n>0. - _R. J. Mathar_, Mar 09 2009
%F E.g.f.: 1 - 8*exp(x)*x^2*(3 + x)/3. - _Stefano Spezia_, Oct 11 2023
%t CoefficientList[Series[(1+x)^2(x^2-6x+1)/(x-1)^4,{x,0,40}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{1,0,-16,-64,-160},40] (* _Harvey P. Dale_, Mar 15 2020 *)
%o (PARI) Vec((1+x)^2*(x^2-6*x+1)/(x-1)^4 + O(x^100)) \\ _Altug Alkan_, Oct 26 2015
%Y Essentially the same as A102860. Cf. A115046, A002928.
%K sign,easy
%O 0,3
%A _Roger L. Bagula_, Apr 07 2008
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