A260784 - OEIS

A260784

Coefficients in a certain low-temperature series expansion.

%I #21 May 03 2024 06:05:51

%S 0,24,1440,181440,43545600,17882726400,11333177856000,

%T 10257397742592000,12540115964952576000,19887027595237490688000,

%U 39679473692005106319360000,97249082487667949725286400000,287164491478121796028858368000000,1005464789964467723115455053824000000

%N Coefficients in a certain low-temperature series expansion.

%H Vaclav Kotesovec, <a href="/A260784/b260784.txt">Table of n, a(n) for n = 1..196</a>

%H Grzegorz Siudem, Agata Fronczak, <a href="https://arxiv.org/abs/2007.16132">Bell polynomials in the series expansions of the Ising model</a>, arXiv:2007.16132 [math-ph], 2020.

%H G. Siudem, A. Fronczak, P. Fronczak, <a href="http://arxiv.org/abs/1410.7963">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</a>, arXiv preprint arXiv:1410.7963, 2014. See equations (8) and (11).

%F 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

%p A260784 := proc(n)

%p local a,d1,d2,d3,d4,d33half ;

%p a := 0 ;

%p for d2 from 0 do

%p if 2*d2 > n then

%p break;

%p end if;

%p for d3 from 0 do

%p if 2*d2 +3*d3 > n then

%p break;

%p end if;

%p for d4 from 0 do

%p if 2*d2 +3*d3+4*d4 > n then

%p break;

%p end if;

%p d1 := n-2*d2-3*d3-4*d4 ;

%p if d1 >= 0 and type(d1+d3,'even') then

%p d13half := (d1+d3)/2 ;

%p a := a+(d1+d2+d3+d4)!/d1!/d2!/d3!/d4!*(-1)^(d2+d3+d4-1)*2^d2

%p /(d1+d2+d3+d4)*binomial(d1+d3,d13half)^2 ;

%p end if;

%p end do:

%p a*n!/2 ;

%p end proc:

%p seq(A260784(2*n),n=1..15) ; # _R. J. Mathar_, Aug 27 2015

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

%Y Cf. A002890.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Aug 04 2015