A117256 - OEIS

%I #5 Mar 30 2012 18:36:56

%S 1,1,1,-10,5,1,750,-250,25,1,-328125,93750,-6250,125,1,779296875,

%T -205078125,11718750,-156250,625,1,-9741210937500,2435302734375,

%U -128173828125,1464843750,-3906250,3125,1,630569458007812500,-152206420898437500,7610321044921875

%N Triangle T, read by rows, where matrix power T^5 has powers of 5 in the secondary diagonal: [T^5](n+1,n) = 5^(n+1), with all 1's in the main diagonal and zeros elsewhere.

%C More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=5, q=5, r=1.

%F T(n,k) = A117257(n-k)*5^((n-k)*k). T(n,k) = (-1)^(n-k)*5^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(5*j-1) for n>k>=0, with T(n,n) = 1.

%e Triangle T begins:

%e 1;

%e 1,1;

%e -10,5,1;

%e 750,-250,25,1;

%e -328125,93750,-6250,125,1;

%e 779296875,-205078125,11718750,-156250,625,1;

%e -9741210937500,2435302734375,-128173828125,1464843750,-3906250,3125,1;

%e Matrix power T^5 has powers of 5 in the 2nd diagonal:

%e 1;

%e 5,1;

%e 0,25,1;

%e 0,0,125,1;

%e 0,0,0,625,1;

%e 0,0,0,0,3125,1; ...

%o (PARI) {T(n,k)=local(m=1,p=5,q=5,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

%Y Cf. A117257 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

%K sign,tabl

%O 0,4

%A _Paul D. Hanna_, Mar 14 2006