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

0,4

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=-2, q=2, r=1.

Links

Table of n, a(n) for n=0..41.

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.

Example

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; ...

Prog

(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)}

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).

Sequence in context: A117269 A291080 A107862 * A107727 A346743 A087041

Adjacent sequences: A117262 A117263 A117264 * A117266 A117267 A117268

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved