A117260 Triangle T, read by rows, where matrix inverse T^-1 has -2^n in the secondary diagonal: [T^-1](n+1,n) = -2^n, with all 1's in the main diagonal and zeros elsewhere.

1, 1, 1, 2, 2, 1, 8, 8, 4, 1, 64, 64, 32, 8, 1, 1024, 1024, 512, 128, 16, 1, 32768, 32768, 16384, 4096, 512, 32, 1, 2097152, 2097152, 1048576, 262144, 32768, 2048, 64, 1, 268435456, 268435456, 134217728, 33554432, 4194304, 262144, 8192, 128, 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=2, r=1.

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

Triangle T begins:
1;
1,1;
2,2,1;
8,8,4,1;
64,64,32,8,1;
1024,1024,512,128,16,1;
32768,32768,16384,4096,512,32,1;
2097152,2097152,1048576,262144,32768,2048,64,1;
268435456,268435456,134217728,33554432,4194304,262144,8192,128,1;
Matrix inverse T^-1 has -2^n in the 2nd diagonal:
1,
-1,1,
0,-2,1,
0,0,-4,1,
0,0,0,-8,1,
0,0,0,0,-16,1,
0,0,0,0,0,-32,1, ...

Mathematica

Flatten[Table[2^((n(n-1))/2-(k(k-1))/2), {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Sep 19 2013 *)

Prog

(PARI) {T(n, k)=local(m=1, p=-1, 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. A006125 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Sequence in context: A135080 A348724 A238182 * A077944 A077992 A126586

Adjacent sequences: A117257 A117258 A117259 * A117261 A117262 A117263

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 14 2006

Status

approved