A370953 Numerators of coefficients of the partition function per spin, lambda (divided by 2), in the very high temperature region, expressed as a power series in the parameter K^2, for the spin-1/2 Ising model on square lattice.
1, 1, 4, 77, 1009, 101627, 1302779, 2513121979, 11291682179, 1354947005798, 23064317580681848, 20189102649892270054, 776220757551441546419, 641273428219629914673014, 5433381672262390009892530636, 1399751922597075578762073697769
Offset: 0
Links
- Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev. 60 (1941), 252-262.
- Hendrik A. Kramers and Gregory H. Wannier. Statistics of the two-dimensional ferromagnet. Part II. Phys. Rev. 60 (1941), 263-276. See (41), p. 263.
- Hendrik A. Kramers and Gregory H. Wannier, Extract from page 263 of Part II.
- Gandhimohan M. Viswanathan, The hypergeometric series for the partition function of the 2D Ising model, J. Stat. Mech. (2015) P07004; arXiv:1411.2495 [cond-mat.stat-mech], 2014-2015.
- Wikipedia, Square lattice Ising model.
Programs
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Mathematica
CoefficientList[With[{nmax = 7}, Exp[-Log[2]/2 + 1/(2 Pi) Integrate[Log[Cosh[2k]^2 + Sqrt[Sinh[2k]^4 + 1 - 2 Sinh[2k]^2 Cos[2\[Theta]] + O[k]^(2nmax+1)]], {\[Theta], 0, Pi}] + O[k]^(2nmax+1)]], k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 10 2024 *) CoefficientList[Cosh[2k] Exp[-x HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16x] /. {x -> (Sinh[2k]/(2Cosh[2k]^2))^2}] + O[k]^32, k][[;; ;; 2]] // Numerator (* Andrey Zabolotskiy, Mar 13 2024, using the g. f. from Gandhimohan M. Viswanathan *)
Formula
a(n) / A370954(n) ~ c * 2^(2*n) / (n^3 * log(1 + sqrt(2))^(2*n)), where c = 0.15662885... - Vaclav Kotesovec, May 02 2024
Extensions
Terms a(5) and beyond from Andrey Zabolotskiy, Mar 10 2024