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.
T(n,k) is the number of simple labeled graphs G on [n] such that the subgraph of G induced by the vertices labeled 1,2,...,k is a clique of size k. Cf A277219. - Geoffrey Critzer, May 05 2024
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Mar 14 2006
STATUS
approved