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A002890
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Low temperature series for spin-1/2 Ising free energy on 2D square lattice.
(Formerly M1463 N0578)
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1
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1, 0, 1, 2, 5, 14, 44, 152, 566, 2234, 9228, 39520, 174271, 787246, 3628992, 17019374, 81011889, 390633382, 1905134695, 9385453576, 46653815395, 233788460256, 1180111379105, 5996452414310, 30653752894948
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OFFSET
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0,4
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COMMENTS
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The g.f. (1-6*z+9*z**2-4*z**3+z**4)/(4*z-1)/(2*z-1) conjectured by Simon Plouffe in his 1992 dissertation is wrong.
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REFERENCES
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P. D. Beale, Exact distribution of energies in the two-dimensional Ising model, Phys. Rev. Lett. 76 (1996) 78-81
C. Domb, On the theory of cooperative phenomena in crystals, Advances in Phys., 9 (1960), 149-361.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=0..24.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. R. Finch, Lenz-Ising Constants
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MATHEMATICA
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(* 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 *)
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CROSSREFS
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Sequence in context: A207081 A148336 A119021 * A202856 A118929 A204064
Adjacent sequences: A002887 A002888 A002889 * A002891 A002892 A002893
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Corrections and updates from Steven Finch
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STATUS
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approved
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