降维法快速求解A(n,k)精确公式

A Rapid Method of Getting the Accurate Formula of A(n,k) by Reduced Dimensions

A(n,k)=sum from m=1 to k sum r=1 to m sum j=0 to [k/m]-1 (tm,r,j (k)×nj×s(r,m)×ζmnr,ζm=e2πi/m,s(r,m)={1,gcd(r,m)=1 0,其他)为丢番图方程sum i=1 to k (ixi=n)的非负整数解的个数.虽然用解线性方程组的方法可求得A(n,k)的所有系数,然而,该求解过程却非常耗时.本文利用方程(1-x)(1-x2)...(1-xk)=0的相异根的幂可能存在的相等关系,即取适当的正整数g使某些相异根的g次幂相等来实现同类项系数的合并以降低方程的维数,达到提高方程求解速度的目的.

Let A （n, k） be the number of nonnegative integer solutions for the Diophantineequation ∑i=1^kixi=n.We can get all coefficients of the A （n, k）, by solving the system of linear equations for the coefficients of the A（n,k）, where, A（n,k） =k↑∑↑m=1m↑∑↑r=1 [k/m]-1↑∑↑j=0tm,rj^（k）×n^j×s（r,m）×ζm^nr,ζm=e^2π/m,s（r,m）={1,gcd（r,m）=1 0,其他. But this processing will costour much time. To solve the problem, this paper has provided a new method, which reduces the dimensions of the system of linear equations by collating coefficients of the qi^g ,qc≠i^g in the A（n,k） when qi^g =qc≠i^g, the roots of the equation（1 -x）（1 -x^2）.. （1 -x^k） =0.Then the dimensions of the constructed system of linear equations are reduced, so improves the speed of solving the equations.