Some Sum-Of-Sum Congruences

[Ralf Stephan]

> qqquet at mindspring.com (Leroy Quet) wrote in message 
> For n>= 2,
> 
> A(n,m) =
> 
> (1/(n-1)!)*
>    sum{k>=0} sum{j=1 to n-1} S(n-1,j)*(
>        [m+1-6k]^j +[m-6k]^j -[m-2-6k]^j -[m-3-6k]^j),
> 
> where S() is an unsigned Stirling number of the first kind,
> and [x] =  maximum(0,x).
> 
> I should also probably give the first few terms of A(n,m):
> 
> A(0,m):  1, 1, 0, -1, -1,  0,  1,  1,  0, -1, -1, 0,...
> A(1,m):  1, 2, 2,  1,  0,  0,  1,  2,  2,  1,  0, 0,...
> A(2,m):  1, 3, 5,  6,  6,  6,  7,  9, 11, 12, 12, 12,...
> A(3,m):  1, 4, 9, 15, 21, 27, 34, 43, 54, 66, 78, 90,...

A(n+1,m) are partial sums of A(n,m), and the first three
are A010892, A021823, A077859.


ralf