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%I #5 Mar 30 2012 18:36:56
%S 1,1,1,-2,4,1,32,-32,16,1,-2560,2048,-512,64,1,917504,-655360,131072,
%T -8192,256,1,-1409286144,939524096,-167772160,8388608,-131072,1024,1,
%U 9070970929152,-5772436045824,962072674304,-42949672960,536870912,-2097152,4096,1
%N Triangle T, read by rows, where matrix power T^2 has 2*4^n in the secondary diagonal: [T^2](n+1,n) = 2*4^n, 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=4, r=1.
%F T(n,k) = A117259(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}(2*j-1) for n>k>=0, with T(n,n) = 1.
%e Triangle T begins:
%e 1;
%e 1,1;
%e -2,4,1;
%e 32,-32,16,1;
%e -2560,2048,-512,64,1;
%e 917504,-655360,131072,-8192,256,1;
%e -1409286144,939524096,-167772160,8388608,-131072,1024,1;
%e Matrix square T^2 has 2*4^n in the 2nd diagonal:
%e 1,
%e 2,1,
%e 0,8,1,
%e 0,0,32,1,
%e 0,0,0,128,1,
%e 0,0,0,0,512,1, ...
%o (PARI) {T(n,k)=local(m=1,p=2,q=4,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
%Y Cf. A117259 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (q=5), 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
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