A277219 -id:A277219 - cp's OEIS frontend

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

Paul D. Hanna, Mar 14 2006

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

			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,
  ...

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).

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

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

{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)}

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

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula