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A117254 Triangle T, read by rows, where matrix power T^4 has powers of 4 in the secondary diagonal: [T^4](n+1,n) = 4^(n+1), with all 1's in the main diagonal and zeros elsewhere.

%I A117254 #5 Mar 30 2012 18:36:55
%S A117254 1,1,1,-6,4,1,224,-96,16,1,-39424,14336,-1536,64,1,30277632,-10092544,
%T A117254 917504,-24576,256,1,-98180268032,31004295168,-2583691264,58720256,
%U A117254 -393216,1024,1,1321338098679808,-402146377859072,31748398252032,-661424963584,3758096384,-6291456,4096,1
%N A117254 Triangle T, read by rows, where matrix power T^4 has powers of 4 in the secondary diagonal: [T^4](n+1,n) = 4^(n+1), with all 1's in the main diagonal and zeros elsewhere.
%C A117254 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=4, q=4, r=1.
%F A117254 T(n,k) = A117255(n-k)*4^((n-k)*k). T(n,k) = (-1)^(n-k)*4^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(4*j-1) for n>k>=0, with T(n,n) = 1.
%e A117254 Triangle T begins:
%e A117254 1;
%e A117254 1,1;
%e A117254 -6,4,1;
%e A117254 224,-96,16,1;
%e A117254 -39424,14336,-1536,64,1;
%e A117254 30277632,-10092544,917504,-24576,256,1;
%e A117254 -98180268032,31004295168,-2583691264,58720256,-393216,1024,1; ...
%e A117254 Matrix power T^4 has powers of 4 in the 2nd diagonal:
%e A117254 1;
%e A117254 4,1;
%e A117254 0,16,1;
%e A117254 0,0,64,1;
%e A117254 0,0,0,256,1;
%e A117254 0,0,0,0,1024,1;
%e A117254 0,0,0,0,0,4096,1; ...
%o A117254 (PARI) {T(n,k)=local(m=1,p=4,q=4,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
%Y A117254 Cf. A117255 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).
%K A117254 sign,tabl
%O A117254 0,4
%A A117254 _Paul D. Hanna_, Mar 14 2006