|
%I M1463 N0578 #58 Feb 12 2022 13:32:43
%S 1,0,1,2,5,14,44,152,566,2234,9228,39520,174271,787246,3628992,
%T 17019374,81011889,390633382,1905134695,9385453576,46653815395,
%U 233788460256,1180111379105,5996452414310,30653752894948
%N Low temperature series for spin-1/2 Ising partition function on 2D square lattice.
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrey Zabolotskiy, <a href="/A002890/b002890.txt">Table of n, a(n) for n = 0..500</a>
%H P. D. Beale, <a href="http://dx.doi.org/10.1103/PhysRevLett.76.78">Exact distribution of energies in the two-dimensional Ising model</a>, Phys. Rev. Lett. 76 (1996) 78-81
%H C. Domb, <a href="http://dx.doi.org/10.1080/00018736000101199">On the theory of cooperative phenomena in crystals</a>, Advances in Phys., 9 (1960), 149-361.
%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/ising/ising.html">Lenz-Ising Constants</a> [broken link]
%H Steven R. Finch, <a href="http://web.archive.org/web/20010207201511/http://www.mathsoft.com:80/asolve/constant/ising/ising.html">Lenz-Ising Constants</a> [From the Wayback Machine]
%H G. Siudem, A. Fronczak, and P. Fronczak, <a href="http://arxiv.org/abs/1410.7963">Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s= 1/2 Ising model on the infinite square lattice</a>, arXiv preprint arXiv:1410.7963 [math-ph], 2014-2015.
%H Gandhimohan M. Viswanathan, <a href="http://arxiv.org/abs/1411.2495">The hypergeometric series for the partition function of the 2-D Ising model</a> arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
%H Gandhimohan M. Viswanathan, <a href="https://arxiv.org/abs/2104.03430">The double hypergeometric series for the partition function of the 2D anisotropic Ising model</a>, arXiv:2104.03430 [cond-mat.stat-mech], 2021.
%t (* For 25 terms, a PC computation lasts less than half an hour *) m = 48 (* max y exponent *); coes = CoefficientList[ Series[ Log[(1 + y^2)^2 - 2*y*(1 - y^2)*(Cos[2*Pi*u] + Cos[2*Pi*v])], {y, 0, m}], y] // Rest; nint[f_, {n_}] := If[n == 2 || OddQ[n], 0, Print[n]; Integrate[ Integrate[f, {u, 0, 1}], {v, 0, 1}]]; fy = MapIndexed[nint, coes].Table[y^k, {k, 1, m}]; CoefficientList[ Series[ Exp[fy/2], {y, 0, m}] , y^2] (* _Jean-François Alcover_, Mar 19 2013 *)
%t CoefficientList[(1+u) Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (u (1 - u)^2)/(1 + u)^4}] + O[u]^50, u] (* _Andrey Zabolotskiy_, Feb 12 2022, using the g. f. from Gandhimohan M. Viswanathan, 2014-2015 *)
%Y Cf. A002891.
%K nonn
%O 0,4
%A _N. J. A. Sloane_
%E Corrections and updates from _Steven Finch_
%E "Free energy" changed back to "partition function" (basically the exponential of the free energy) in the name by _Andrey Zabolotskiy_, Feb 11 2022
|
