A117252 - OEIS

%I #5 Mar 30 2012 18:36:55

%S 1,1,1,-3,3,1,45,-27,9,1,-2430,1215,-243,27,1,433026,-196830,32805,

%T -2187,81,1,-245525742,105225318,-15943230,885735,-19683,243,1,

%U 434685788658,-178988265918,25569752274,-1291401630,23914845,-177147,729,1

%N Triangle T, read by rows, where matrix power T^3 has powers of 3 in the secondary diagonal: [T^3](n+1,n) = 3^(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=3, q=3, r=1.

%F T(n,k) = A117253(n-k)*3^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-3*j)]/(n-k)!*3^(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 -3,3,1;

%e 45,-27,9,1;

%e -2430,1215,-243,27,1;

%e 433026,-196830,32805,-2187,81,1;

%e -245525742,105225318,-15943230,885735,-19683,243,1;

%e 434685788658,-178988265918,25569752274,-1291401630,23914845,-177147,729,1;

%e Matrix cube T^3 has powers of 3 in the 2nd diagonal:

%e 1;

%e 3,1;

%e 0,9,1;

%e 0,0,27,1;

%e 0,0,0,81,1;

%e 0,0,0,0,243,1;

%e 0,0,0,0,0,729,1; ...

%o (PARI) {T(n,k)=local(m=1,p=3,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. A117253 (column 0); variants: A117250 (p=q=2), 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 0,4

%A _Paul D. Hanna_, Mar 14 2006