A010573 -id:A010573 - 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.

2, 4, 8, 24, 84, 328, 1372, 6024, 27412, 128228, 613160, 2985116, 14751592, 73825416, 373488764, 1907334616, 9820757380, 50934592820, 265877371160, 1395907472968, 7366966846564, 39062802311672, 208015460898924, 1112050252939612, 5966352507546872

Offset: 1

N. J. A. Sloane, Simon Plouffe

Previous name was: Energy function for square lattice.

C. Domb, Ising model, in Phase Transitions and Critical Phenomena, vol. 3, ed. C. Domb and M. S. Green, Academic Press, 1974; p. 386.
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).

C. Domb, Ising model, Phase Transitions and Critical Phenomena 3 (1974), 257, 380-381, 384-387, 390-391, 412-423. (Annotated scanned copy)
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
Lars Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Phys. Rev. 65, 117 (1944).
M. F. Sykes and M. E. Fisher, Antiferromagnetic susceptibility of the plane square and honeycomb Ising lattices, Physica, 28 (1962), 919-938.

Cf. A002906-A002930, A010571, A010572, A010573, A010574.

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

u[h_]:=Coth[2h](1+(2/Pi)(2Tanh[2h]^2-1)EllipticK[(2Sinh[2h]/Cosh[2h]^2)^2]);
Table[SeriesCoefficient[u[ArcTanh[v]],{v,0,2n-1}],{n,10}]
(* Andrey Zabolotskiy, Sep 12 2017; see Onsager's eq. (116) *)
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 *)

a(n) ~ 2 * (1 + sqrt(2))^(2*n-1) / (Pi * n^2). - Vaclav Kotesovec, Apr 27 2024

More terms and new name from Andrey Zabolotskiy, Oct 19 2017

A010572 High-temperature coefficients for the internal energy for spin-1/2 Ising model on 4-d cubic lattice.

Original entry on oeis.org

4, 24, 432, 10512, 290552, 8800432, 284289160, 9621738448

Offset: 0

N. J. A. Sloane

M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and "high-density" expansions, Phys. Rev. 133 (1964) A224-A239.

Cf. A010571 (3D), A010573 (5D), A010574 (6D), A030044 (partition function), A010557 (fourth-field derivative of free energy), A010563.

Sum_{n>=0} a(n) * v^(2*n+1) = v*q/2 + (1-v^2) * f'(v) / f(v), where f(v) = Sum_{n>=0} A030044(n) * v^(2*n) and q = 8 is the number of nearest neighbors. - Andrey Zabolotskiy, Feb 16 2022

a(5)-a(7) from Andrey Zabolotskiy, Feb 16 2022, corrected Nov 26 2024

A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

Original entry on oeis.org

1, 0, 10, 180, 5075, 180496

Offset: 0

Steven Finch

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 391-406.

Daniel Andrén, Series expansion for the density of states of the Ising and Potts models, arXiv:0706.3116 [cond-mat.str-el], 2007.
Steven R. Finch, Lenz-Ising Constants [broken link]
Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and "high-density" expansions, Phys. Rev. 133 (1964) A224-A239. See p. A226, in particular Eq. (2.11) (together with Eq. (4.10)), for the power series with coefficients g_n, which is the logarithm of this power series, and its physical and combinatorial interpretations.

Cf. A010579 (susceptibility), A010573 (internal energy), A001393 (3D cubic lattice), A030044 (4D cubic lattice).

"Free energy" corrected to "partition function" (basically the exponential of the free energy) in the name by Andrey Zabolotskiy, Feb 12 2022

a(5) added by Andrey Zabolotskiy, Jun 30 2022 using Andrén's data (see his Table 4, column b_n^5 for the coefficients of the expansion of the logarithm of the g.f. of this sequence)

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.

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A010572 High-temperature coefficients for the internal energy for spin-1/2 Ising model on 4-d cubic lattice.

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A030048 High temperature series for spin-1/2 Ising partition function on 5D simple cubic lattice.

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