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

0,1

References

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

Links

Table of n, a(n) for n=0..22.

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

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

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 *)

Crossrefs

Cf. A002890 (partition function).

Equals A029873/4 or A029874*8.

Sequence in context: A212513 A279905 A146748 * A056633 A248464 A232572

Adjacent sequences: A029869 A029870 A029871 * A029873 A029874 A029875

Keyword

nonn

Author

Steven Finch

Extensions

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

Status

approved