A117262 - OEIS

%I #4 Mar 30 2012 18:36:56

%S 1,1,1,3,3,1,27,27,9,1,729,729,243,27,1,59049,59049,19683,2187,81,1,

%T 14348907,14348907,4782969,531441,19683,243,1,10460353203,10460353203,

%U 3486784401,387420489,14348907,177147,729,1

%N 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 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 T(n,k) = 3^(n*(n-1)/2 - k*(k-1)/2).

%e Triangle T begins:

%e 1;

%e 1,1;

%e 3,3,1;

%e 27,27,9,1;

%e 729,729,243,27,1;

%e 59049,59049,19683,2187,81,1;

%e 14348907,14348907,4782969,531441,19683,243,1;

%e 10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;

%e Matrix inverse T^-1 has -3^n in the 2nd diagonal:

%e 1,

%e -1,1,

%e 0,-3,1,

%e 0,0,-9,1,

%e 0,0,0,-27,1,

%e 0,0,0,0,-81,1,

%e 0,0,0,0,0,-243,1, ...

%o (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 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 nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Mar 14 2006