A054389 Susceptibility series H_5 for 2-dimensional Ising model (divided by 2).

1, 20, 140, 620, 2016, 5364, 12292, 25228, 47488, 83508, 138908, 220748, 337568, 499668, 719124, 1010092, 1388800, 1873876, 2486316, 3249836, 4190816, 5338676, 6725796, 8387916, 10364032, 12696820, 15432508, 18621324, 22317344, 26578964, 31468724, 37053804

Offset

0,2

Links

Colin Barker, Table of n, a(n) for n = 0..1000

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 (4,-4,-4,10,-4,-4,4,-1).

Formula

G.f.: (1 + 16*x + 64*x^2 + 144*x^3 + 166*x^4 + 144*x^5 + 64*x^6 + 16*x^7 + x^8) / ((1 - x)^6*(1 + x)^2).

From Colin Barker, Dec 09 2016: (Start)

a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n>8.

a(n) = (77*n^5 + 630*n^3 + 448*n)/60 for n>0 and even.

a(n) = (77*n^5 + 630*n^3 + 493*n)/60 for n odd. (End)

From G. C. Greubel, Jul 31 2019: (Start)

a(n) = n*(154*n^4 + 1260*n^2 + 941 - 45*(-1)^n)/120, n>0, with a(0)=1.

E.g.f.: (x*(2355 + 6090*x + 5110*x^2 + 1540*x^3 + 154*x^4)*exp(x) + 120 + 45*x*exp(-x))/120. (End)

Mathematica

LinearRecurrence[{4, -4, -4, 10, -4, -4, 4, -1}, {1, 20, 140, 620, 2016, 5364, 12292, 25228, 47488}, 35] (* or *) CoefficientList[Series[(1 +16*x +64*x^2 + 144*x^3 +166*x^4 +144*x^5 +64*x^6 +16*x^7 +x^8)/((1-x)^6*(1+x)^2), {x, 0, 35}], x] (* Indranil Ghosh, Feb 24 2017 *)
Table[If[n==0, 1, n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120], {n, 0, 35}] (* G. C. Greubel, Jul 31 2019 *)

Prog

(PARI) Vec((1 +16*x +64*x^2 +144*x^3 +166*x^4 +144*x^5 +64*x^6 +16*x^7 + x^8)/((1-x)^6*(1+x)^2) + O(x^35)) \\ Colin Barker, Dec 09 2016
(Magma) [1] cat [n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120: n in [1..35]]; // G. C. Greubel, Jul 31 2019
(Sage) [1]+[n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120 for n in (1..35)] # G. C. Greubel, Jul 31 2019
(GAP) Concatenation([1], List([1..35], n-> n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120)); # G. C. Greubel, Jul 31 2019

Crossrefs

Cf. A008574, A054275, A054410, A054764.

Sequence in context: A358865 A134382 A105939 * A253003 A293932 A071816

Adjacent sequences: A054386 A054387 A054388 * A054390 A054391 A054392

Keyword

nonn,easy

Author

N. J. A. Sloane, May 09 2000

Status

approved