A029872 Low temperature series for spin-1/2 Ising specific heat on 2D square lattice.

16, 72, 288, 1200, 5376, 25480, 125504, 634608, 3269680, 17086168, 90282240, 481347152, 2585485504, 13974825960, 75941188736, 414593263952, 2272626444528, 12502223573304, 68996534259040, 381858968527680, 2118806030647328, 11783826597027256, 65674579024955904

Offset: 0

Author

Steven Finch

Keywords

• nonn

References

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

Links

• G. A. Baker, Further application of the Padé approximant method to the Ising and Heisenberg models, Phys. Rev. 129 (1963) 99-102.
• I. G. Enting, A, J. Guttmann and I. Jensen, Low-Temperature Series Expansions for the Spin-1 Ising Model, arXiv:hep-lat/9410005, 1994; J. Phys. A. 27 (1994) 6987-7006.
• Steven R. Finch, Lenz-Ising Constants [broken link]
• Steven R. Finch, Lenz-Ising Constants [From the Wayback Machine]
• Index entries for sequences related to specific heat

Crossrefs

Cf. A002890 (partition function).

Equals A029873/4 or A029874*8.

Programs

Mathematica

CoefficientList[Series[1/(Pi*x^2*(-1 + x^2)^2) * (-2*Pi*x*(1 + x)^2 - (1 + x)^4 * EllipticE[16*(-1 + x)^2*x/(1 + x)^4] + (1 + 30*x^2 + x^4) * EllipticK[16*(-1 + x)^2*x/(1 + x)^4]), {x, 0, 25}], x] (* Vaclav Kotesovec, Apr 28 2024 *)

Formula

G.f.: ((u^4 + 30*u^2 + 1) * K(x) / Pi - (u+1)^4 * E(x) / Pi - 2*u*(u+1)^2) / (u^2 * (u^2-1)^2) = 4 * (f(u) * (f'(u)/u + f''(u)) - (f'(u))^2) / f(u)^2, where f(u) is the g.f. of A002890, K(x) and E(x) are the complete elliptic integrals, x = 4*(1-u)*sqrt(u)/(1+u)^2. - Andrey Zabolotskiy, Feb 15 2022

a(n) ~ 2 * (1 + sqrt(2))^(2*n+4) / (Pi*n). - Vaclav Kotesovec, Apr 28 2024

Extensions

Terms a(18) and beyond from Andrey Zabolotskiy, Feb 15 2022