|
%I #21 Sep 08 2022 08:45:00
%S 1,20,140,620,2016,5364,12292,25228,47488,83508,138908,220748,337568,
%T 499668,719124,1010092,1388800,1873876,2486316,3249836,4190816,
%U 5338676,6725796,8387916,10364032,12696820,15432508,18621324,22317344,26578964,31468724,37053804
%N Susceptibility series H_5 for 2-dimensional Ising model (divided by 2).
%H Colin Barker, <a href="/A054389/b054389.txt">Table of n, a(n) for n = 0..1000</a>
%H A. J. Guttmann, <a href="https://doi.org/10.1016/S0012-365X(99)00262-9">Indicators of solvability for lattice models</a>, Discrete Math., 217 (2000), 167-189.
%H D. Hansel et al., <a href="http://dx.doi.org/10.1007/BF01010400">Analytical properties of the anisotropic cubic Ising model</a>, J. Stat. Phys., 48 (1987), 69-80.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,-4,10,-4,-4,4,-1).
%F 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).
%F From _Colin Barker_, Dec 09 2016: (Start)
%F 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.
%F a(n) = (77*n^5 + 630*n^3 + 448*n)/60 for n>0 and even.
%F a(n) = (77*n^5 + 630*n^3 + 493*n)/60 for n odd. (End)
%F From _G. C. Greubel_, Jul 31 2019: (Start)
%F a(n) = n*(154*n^4 + 1260*n^2 + 941 - 45*(-1)^n)/120, n>0, with a(0)=1.
%F 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)
%t 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 *)
%t 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 *)
%o (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
%o (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
%o (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
%o (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
%Y Cf. A008574, A054275, A054410, A054764.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, May 09 2000
|
