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

0,2

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).

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

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

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 *)

Prog

(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
(Magma) R<x>:=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
(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

Crossrefs

Cf. A054275, A054410, A054389, A054764, A055857.

Sequence in context: A253131 A119771 A055920 * A195591 A240292 A050406

Adjacent sequences: A055853 A055854 A055855 * A055857 A055858 A055859

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 07 2000

Status

approved