A260784 Coefficients in a certain low-temperature series expansion.

0, 24, 1440, 181440, 43545600, 17882726400, 11333177856000, 10257397742592000, 12540115964952576000, 19887027595237490688000, 39679473692005106319360000, 97249082487667949725286400000, 287164491478121796028858368000000, 1005464789964467723115455053824000000

Offset: 1

Author

N. J. A. Sloane, Aug 04 2015

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..196
• Grzegorz Siudem, Agata Fronczak, Bell polynomials in the series expansions of the Ising model, arXiv:2007.16132 [math-ph], 2020.
• G. Siudem, A. Fronczak, P. Fronczak, Exact low-temperature series expansion for the partition function of the two-dimensional zero-field s= 1/2 Ising model on the infinite square lattice, arXiv preprint arXiv:1410.7963, 2014. See equations (8) and (11).

Crossrefs

Cf. A002890.

Programs

Maple

A260784 := proc(n)
    local a,d1,d2,d3,d4,d33half ;
    a := 0 ;
    for d2 from 0 do
        if 2*d2 > n then
            break;
        end if;
        for d3 from 0 do
            if 2*d2 +3*d3 > n then
                break;
            end if;
            for d4 from 0 do
                if 2*d2 +3*d3+4*d4 > n then
                    break;
                end if;
                d1 := n-2*d2-3*d3-4*d4 ;
                if d1 >= 0 and type(d1+d3,'even') then
                    d13half := (d1+d3)/2 ;
                    a := a+(d1+d2+d3+d4)!/d1!/d2!/d3!/d4!*(-1)^(d2+d3+d4-1)*2^d2
                        /(d1+d2+d3+d4)*binomial(d1+d3,d13half)^2 ;
                end if;
            end do:
        end do:
    end do:
    a*n!/2 ;
end proc:
seq(A260784(2*n),n=1..15) ; # R. J. Mathar, Aug 27 2015

Mathematica

"Listing 1" in Siudem et al. (2014) gives Mathematica code for the fractions a(n)/(2n)!.

Formula

a(n) ~ 2^(2*n) * (1 + sqrt(2))^(2*n) * n^(2*n - 5/2) / (sqrt(Pi) * exp(2*n)). - Vaclav Kotesovec, May 03 2024