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A117262 Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal and zeros elsewhere.

%I A117262 #4 Mar 30 2012 18:36:56
%S A117262 1,1,1,3,3,1,27,27,9,1,729,729,243,27,1,59049,59049,19683,2187,81,1,
%T A117262 14348907,14348907,4782969,531441,19683,243,1,10460353203,10460353203,
%U A117262 3486784401,387420489,14348907,177147,729,1
%N A117262 Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: [T^-1](n+1,n) = -3^n, with all 1's in the main diagonal and zeros elsewhere.
%C A117262 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=-1, q=3, r=1.
%F A117262 T(n,k) = 3^(n*(n-1)/2 - k*(k-1)/2).
%e A117262 Triangle T begins:
%e A117262 1;
%e A117262 1,1;
%e A117262 3,3,1;
%e A117262 27,27,9,1;
%e A117262 729,729,243,27,1;
%e A117262 59049,59049,19683,2187,81,1;
%e A117262 14348907,14348907,4782969,531441,19683,243,1;
%e A117262 10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
%e A117262 Matrix inverse T^-1 has -3^n in the 2nd diagonal:
%e A117262 1,
%e A117262 -1,1,
%e A117262 0,-3,1,
%e A117262 0,0,-9,1,
%e A117262 0,0,0,-27,1,
%e A117262 0,0,0,0,-81,1,
%e A117262 0,0,0,0,0,-243,1, ...
%o A117262 (PARI) {T(n,k)=local(m=1,p=-1,q=3,r=1);prod(j=0,n-k-1,m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}
%Y A117262 Cf. A047656 (column 0), A117263 (row sums); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117265 (p=-2, q=2).
%K A117262 nonn,tabl
%O A117262 0,4
%A A117262 _Paul D. Hanna_, Mar 14 2006