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.

1, 1, 1, 3, 3, 1, 27, 27, 9, 1, 729, 729, 243, 27, 1, 59049, 59049, 19683, 2187, 81, 1, 14348907, 14348907, 4782969, 531441, 19683, 243, 1, 10460353203, 10460353203, 3486784401, 387420489, 14348907, 177147, 729, 1

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..35.

Formula

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

Example

Triangle T begins:
1;
1,1;
3,3,1;
27,27,9,1;
729,729,243,27,1;
59049,59049,19683,2187,81,1;
14348907,14348907,4782969,531441,19683,243,1;
10460353203,10460353203,3486784401,387420489,14348907,177147,729,1;
Matrix inverse T^-1 has -3^n in the 2nd diagonal:
1,
-1,1,
0,-3,1,
0,0,-9,1,
0,0,0,-27,1,
0,0,0,0,-81,1,
0,0,0,0,0,-243,1, ...

Prog

(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)}

Crossrefs

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).

Sequence in context: A259876 A276402 A318110 * A065431 A271082 A053375

Adjacent sequences: A117259 A117260 A117261 * A117263 A117264 A117265

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved