A055856 Susceptibility series H_4 for 2-dimensional Ising model (divided by 2).
1, 16, 90, 328, 888, 2016, 3994, 7212, 12070, 19112, 28846, 41976, 59116, 81132, 108738, 142972, 184638, 234952, 294806, 365596, 448296, 544492, 655230, 782292, 926794, 1090716, 1275238, 1482548, 1713880, 1971636, 2257102, 2572896, 2920350, 3302308, 3720138
Offset: 0
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- A. J. Guttmann and I. G. Enting, Solvability of some statistical mechanical systems, Phys. Rev. Lett., 76 (1996), 344-347.
- A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
- D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
- Index entries for linear recurrences with constant coefficients, signature (1,3,-2,-4,0,4,2,-3,-1,1).
Programs
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Magma
R
:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1 +15*x +71*x^2 +192*x^3 +326*x^4 +388*x^5 +326*x^6 +192*x^7 + 71*x^8 +15*x^9 +x^10)/((1-x^3)*(1-x^2)^3*(1-x)) )); // G. C. Greubel, Jan 16 2020 -
Maple
1,seq( simplify( (4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*ChebyshevT(n, -1/2))/1728 ), n=1..40); # G. C. Greubel, Jan 16 2020
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Mathematica
Join[{1}, Table[(4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*ChebyshevT[n, -1/2])/1728, {n,40}]] (* G. C. Greubel, Jan 16 2020 *) LinearRecurrence[{1,3,-2,-4,0,4,2,-3,-1,1},{1,16,90,328,888,2016,3994,7212,12070,19112,28846},40] (* Harvey P. Dale, Jul 24 2021 *) -
PARI
Vec((1 +15*x +71*x^2 +192*x^3 +326*x^4 +388*x^5 +326*x^6 +192*x^7 + 71*x^8 +15*x^9 +x^10)/((1-x^3)*(1-x)^4*(1+x)^3) + O(x^40)) \\ Colin Barker, Dec 10 2016
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Sage
[1]+[(4794*n^4 +19194*n^2 +3349 -81*(-1)^n*(2*n^2 +5) + 512*chebyshev_T(n, -1/2))/1728 for n in (1..40)] # G. C. Greubel, Jan 16 2020
Formula
G.f.: (1 + 15*x + 71*x^2 + 192*x^3 + 326*x^4 + 388*x^5 + 326*x^6 + 192*x^7 + 71*x^8 + 15*x^9 + x^10)/((1-x^3)*(1-x)^4*(1+x)^3).
a(n) = (4794*n^4 + 19194*n^2 + 3349 - 81*(-1)^n*(2*n^2 + 5) + 512*ChebyshevT(n, -1/2))/1728, for n >= 1, with a(0) = 1. - G. C. Greubel, Jan 16 2020