OFFSET
1,2
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).
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
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)!.
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 04 2015
STATUS
approved