A370953 - OEIS

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.

%I #26 May 02 2024 17:05:39

%S 1,1,4,77,1009,101627,1302779,2513121979,11291682179,1354947005798,

%T 23064317580681848,20189102649892270054,776220757551441546419,

%U 641273428219629914673014,5433381672262390009892530636,1399751922597075578762073697769

%N 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.

%H Hendrik A. Kramers and Gregory H. Wannier. <a href="https://doi.org/10.1103/PhysRev.60.252">Statistics of the two-dimensional ferromagnet. Part I</a>. Phys. Rev. 60 (1941), 252-262.

%H Hendrik A. Kramers and Gregory H. Wannier. <a href="https://doi.org/10.1103/PhysRev.60.263">Statistics of the two-dimensional ferromagnet. Part II</a>. Phys. Rev. 60 (1941), 263-276. See (41), p. 263.

%H Hendrik A. Kramers and Gregory H. Wannier, <a href="/A370953/a370953.pdf">Extract from page 263 of Part II.</a>

%H Gandhimohan M. Viswanathan, <a href="https://doi.org/10.1088/1742-5468/2015/07/P07004">The hypergeometric series for the partition function of the 2D Ising model</a>, J. Stat. Mech. (2015) P07004; arXiv:<a href="https://arxiv.org/abs/1411.2495">1411.2495</a> [cond-mat.stat-mech], 2014-2015.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Square_lattice_Ising_model">Square lattice Ising model</a>.

%F 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

%t 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 *)

%t 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 *)

%Y See A370954 for denominators.

%Y Cf. A370955, A002908, A002890.

%K nonn,frac

%O 0,3

%A _N. J. A. Sloane_, Mar 10 2024

%E Terms a(5) and beyond from _Andrey Zabolotskiy_, Mar 10 2024