A136264 Expansion of g.f. (1+x)^2*(x^2-6*x+1)/(x-1)^4.

1, 0, -16, -64, -160, -320, -560, -896, -1344, -1920, -2640, -3520, -4576, -5824, -7280, -8960, -10880, -13056, -15504, -18240, -21280, -24640, -28336, -32384, -36800, -41600, -46800, -52416, -58464, -64960, -71920, -79360, -87296, -95744, -104720, -114240, -124320, -134976, -146224, -158080

Offset: 0

Author

Roger L. Bagula, Apr 07 2008

Keywords

• sign
• easy

Comments

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.

References

• 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.

Links

• M. R. Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Essentially the same as A102860. Cf. A115046, A002928.

Programs

Mathematica

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 *)

PARI

Vec((1+x)^2*(x^2-6*x+1)/(x-1)^4 + O(x^100)) \\ Altug Alkan, Oct 26 2015

Formula

a(n) = 8*n*(1 - n^2)/3, n>0. - R. J. Mathar, Mar 09 2009

E.g.f.: 1 - 8*exp(x)*x^2*(3 + x)/3. - Stefano Spezia, Oct 11 2023