A117265 Triangle T, read by rows, where matrix power T^-2 has -2^(n+1) in the secondary diagonal: [T^-2](n+1,n) = -2^(n+1), with all 1's in the main diagonal and zeros elsewhere.

1, 1, 1, 3, 2, 1, 20, 12, 4, 1, 280, 160, 48, 8, 1, 8064, 4480, 1280, 192, 16, 1, 473088, 258048, 71680, 10240, 768, 32, 1, 56229888, 30277632, 8257536, 1146880, 81920, 3072, 64, 1, 13495173120, 7197425664, 1937768448, 264241152, 18350080, 655360

Offset: 0

Author

Paul D. Hanna, Mar 14 2006

Keywords

• nonn
• 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=2, r=1.

Examples

			Triangle T begins:
1;
1,1;
3,2,1;
20,12,4,1;
280,160,48,8,1;
8064,4480,1280,192,16,1;
473088,258048,71680,10240,768,32,1;
56229888,30277632,8257536,1146880,81920,3072,64,1;
13495173120,7197425664,1937768448,264241152,18350080,655360,12288,128,1;
Matrix inverse square T^-2 has -2^(n+1) in the 2nd diagonal:
1;
-2,1;
0,-4,1;
0,0,-8,1;
0,0,0,-16,1;
0,0,0,0,-32,1;
0,0,0,0,0,-64,1; ...
		

Crossrefs

Cf. A086229 (column 0), A117266 (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), A117262 (p=-1, q=3).

Programs

PARI

{T(n,k)=local(m=1,p=-2,q=2,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) = A086229(n-k)*2^((n-k)*k). T(n,k) = 2^(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.