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%I #4 Mar 30 2012 18:36:55
%S 1,1,1,-1,2,1,4,-4,4,1,-40,32,-16,8,1,896,-640,256,-64,16,1,-43008,
%T 28672,-10240,2048,-256,32,1,4325376,-2752512,917504,-163840,16384,
%U -1024,64,1,-899678208,553648128,-176160768,29360128,-2621440,131072,-4096,128,1
%N Triangle T, read by rows, where matrix power T^2 has powers of 2 in the secondary diagonal: [T^2](n+1,n) = 2^(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=2, q=2, r=1.
%F T(n,k) = A117251(n-k)*2^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-2*j)]/(n-k)!*2^(n*(n-1)/2 - k*(k-1)/2) for n>k>=0, with T(n,n) = 1.
%e Triangle T begins:
%e 1;
%e 1,1;
%e -1,2,1;
%e 4,-4,4,1;
%e -40,32,-16,8,1;
%e 896,-640,256,-64,16,1;
%e -43008,28672,-10240,2048,-256,32,1;
%e 4325376,-2752512,917504,-163840,16384,-1024,64,1;
%e -899678208,553648128,-176160768,29360128,-2621440,131072,-4096,128,1;
%e Matrix square T^2 has powers of 2 in the 2nd diagonal:
%e 1;
%e 2,1;
%e 0,4,1;
%e 0,0,8,1;
%e 0,0,0,16,1;
%e 0,0,0,0,32,1;
%e 0,0,0,0,0,64,1; ...
%o (PARI) {T(n,k)=local(m=1,p=2,q=2,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
%Y Cf. A117251 (column 0); variants: A117252 (p=q=3), A117254 (p=q=4), 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 sign,tabl
%O 0,5
%A _Paul D. Hanna_, Mar 14 2006
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