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

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=-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

Examples

			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,
  ...
		

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).

Programs

Maple

T := (n, k) -> 2^(((n + k - 1)*(n - k))/2):
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Dec 31 2024

Mathematica

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

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)}

Formula

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