cp's OEIS Frontend

A260784 Coefficients in a certain low-temperature series expansion.

%I A260784 #21 May 03 2024 06:05:51
%S A260784 0,24,1440,181440,43545600,17882726400,11333177856000,
%T A260784 10257397742592000,12540115964952576000,19887027595237490688000,
%U A260784 39679473692005106319360000,97249082487667949725286400000,287164491478121796028858368000000,1005464789964467723115455053824000000
%N A260784 Coefficients in a certain low-temperature series expansion.
%H A260784 Vaclav Kotesovec, <a href="/A260784/b260784.txt">Table of n, a(n) for n = 1..196</a>
%H A260784 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 A260784 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 A260784 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 A260784 := proc(n)
%p A260784     local a,d1,d2,d3,d4,d33half ;
%p A260784     a := 0 ;
%p A260784     for d2 from 0 do
%p A260784         if 2*d2 > n then
%p A260784             break;
%p A260784         end if;
%p A260784         for d3 from 0 do
%p A260784             if 2*d2 +3*d3 > n then
%p A260784                 break;
%p A260784             end if;
%p A260784             for d4 from 0 do
%p A260784                 if 2*d2 +3*d3+4*d4 > n then
%p A260784                     break;
%p A260784                 end if;
%p A260784                 d1 := n-2*d2-3*d3-4*d4 ;
%p A260784                 if d1 >= 0 and type(d1+d3,'even') then
%p A260784                     d13half := (d1+d3)/2 ;
%p A260784                     a := a+(d1+d2+d3+d4)!/d1!/d2!/d3!/d4!*(-1)^(d2+d3+d4-1)*2^d2
%p A260784                         /(d1+d2+d3+d4)*binomial(d1+d3,d13half)^2 ;
%p A260784                 end if;
%p A260784             end do:
%p A260784         end do:
%p A260784     end do:
%p A260784     a*n!/2 ;
%p A260784 end proc:
%p A260784 seq(A260784(2*n),n=1..15) ; # _R. J. Mathar_, Aug 27 2015
%t A260784 "Listing 1" in Siudem et al. (2014) gives Mathematica code for the fractions a(n)/(2n)!.
%Y A260784 Cf. A002890.
%K A260784 nonn
%O A260784 1,2
%A A260784 _N. J. A. Sloane_, Aug 04 2015