cp's OEIS Frontend

A002908 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on square lattice with nearest-neighbor interaction J at temperature T.

%I A002908 M1161 N0444 #34 Apr 27 2024 15:53:35
%S A002908 2,4,8,24,84,328,1372,6024,27412,128228,613160,2985116,14751592,
%T A002908 73825416,373488764,1907334616,9820757380,50934592820,265877371160,
%U A002908 1395907472968,7366966846564,39062802311672,208015460898924,1112050252939612,5966352507546872
%N A002908 High temperature expansion of -u/J in odd powers of v = tanh(J/kT), where u is energy per site of the spin-1/2 Ising model on square lattice with nearest-neighbor interaction J at temperature T.
%C A002908 Previous name was: Energy function for square lattice.
%D A002908 C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.
%D A002908 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002908 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002908 C. Domb, <a href="/A007239/a007239.pdf">Ising model</a>, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
%H A002908 M. E. Fisher and D. S. Gaunt, <a href="https://doi.org/10.1103/PhysRev.133.A224">Ising model and self-avoiding walks on hypercubical lattices and high density expansions</a>, Phys. Rev. 133 (1964) A224-A239.
%H A002908 Lars Onsager, <a href="https://doi.org/10.1103/PhysRev.65.117">Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition</a>, Phys. Rev. 65, 117 (1944).
%H A002908 M. F. Sykes and M. E. Fisher, <a href="https://doi.org/10.1016/0031-8914(62)90080-0">Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices</a>, Physica, 28 (1962), 919-938.
%F A002908 a(n) ~ 2 * (1 + sqrt(2))^(2*n-1) / (Pi * n^2). - _Vaclav Kotesovec_, Apr 27 2024
%p A002908 series((1+v^2)*(1-(2/Pi)*(1-6*v^2+v^4)*EllipticK(4*v*(1-v^2)/(1+v^2)^2)/(1+v^2)^2)/2*v,v,50); # _Sean A. Irvine_, Nov 26 2017
%t A002908 u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);
%t A002908 Table[SeriesCoefficient[u[ArcTanh[v]],{v,0,2n-1}],{n,10}]
%t A002908 (* _Andrey Zabolotskiy_, Sep 12 2017; see Onsager's eq. (116) *)
%t A002908 Rest[CoefficientList[Series[(1+x)/2 - (1 - 6*x + x^2)*EllipticK[(16*(-1 + x)^2*x)/(1 + x)^4] / (Pi*(1+x)), {x, 0, 25}], x]] (* _Vaclav Kotesovec_, Apr 27 2024 *)
%Y A002908 Cf. A002906-A002930, A010571, A010572, A010573, A010574.
%K A002908 nonn,nice
%O A002908 1,1
%A A002908 _N. J. A. Sloane_, _Simon Plouffe_
%E A002908 More terms and new name from _Andrey Zabolotskiy_, Oct 19 2017