Series and other data related to the fiveparticle contribution χ^{(6)} to the square lattice Ising model susceptibility.
First a bit of notation. The `natural' variables are s=sinh(2J/kT) and w=s/(2(1+s^{2})) and x=w^{2}.
Some exact results for nparticle contributions to the susceptibility
χ^{(2)} = 4w^{4} _{2}F _{1}[5/2,3/2;3;16w^{2}]  
ODE for χ^{(4)} in variable x=w^{2}.  
First column is the order k of the derivative, the second column is j and the third column is the j'th coefficient p_{k,j} in the polynomial P_{k}(w) multiplying the k'th derivative.  
ODE for χ^{(6)} in variable x mod the prime 32749.  
For χ^{(6)} we express the solution in terms of a linear ODE with polynomial coefficients but using the diffential operator (wd/dw). The solution is of order n=56 with the degree of the polynomials equal 129. The data is organised as a list of lists with the first list being the coefficients of P_{n}(w), the second list being the coefficients of P_{n1}(w) etc. etc. and the last list is the coefficients of P_{0}(w)  
Some exact series mod various primes.

For the prime 32749 the series has some 10000 terms
while for the remaining primes the series has some 5600 terms.
Series for χ^{(4)} in w mod the primes: 32749 32719 32717 32713 
Series for χ^{(6)} in w mod the primes: 32749 32719 32717 32713 
Some exact results for factors occuring in the order 46 differential operator L_{46} annihilating the series Φ^{(6)}=χ^{(6)}2/3χ^{(4)}+2/45χ^{(2)}

First we have the operator L_{17} in exact arithmetic.
This operator has the factorization:
Next we have the operator L_{6} mod 32749. This is the minimal order ODE for this operator.
A solution to this ODE is given by
where E and K denote the complete elliptic integrals
while P_{4i,i} are polynomials in w with coefficients known modulo the prime 32749 and of degree 200, 202, 204, 204 and 204, respectively. Click here to download these polynomials .