A117258 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.

1, 1, 1, -2, 4, 1, 32, -32, 16, 1, -2560, 2048, -512, 64, 1, 917504, -655360, 131072, -8192, 256, 1, -1409286144, 939524096, -167772160, 8388608, -131072, 1024, 1, 9070970929152, -5772436045824, 962072674304, -42949672960, 536870912, -2097152, 4096, 1

Offset: 0

Author

Paul D. Hanna, Mar 14 2006

Keywords

• sign
• tabl

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=2, q=4, r=1.

Examples

			Triangle T begins:
1;
1,1;
-2,4,1;
32,-32,16,1;
-2560,2048,-512,64,1;
917504,-655360,131072,-8192,256,1;
-1409286144,939524096,-167772160,8388608,-131072,1024,1;
Matrix square T^2 has 2*4^n in the 2nd diagonal:
1,
2,1,
0,8,1,
0,0,32,1,
0,0,0,128,1,
0,0,0,0,512,1, ...
		

Crossrefs

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).

Programs

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)}

Formula

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.