A378666 Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.

1, 1, 1, 1, 12, 1, 1, 117, 117, 1, 1, 1080, 10530, 1080, 1, 1, 9801, 882090, 882090, 9801, 1, 1, 88452, 72243171, 666860040, 72243171, 88452, 1, 1, 796797, 5873190687, 491992666011, 491992666011, 5873190687, 796797, 1, 1, 7173360, 476309310660, 360089838858960, 3267815287645062, 360089838858960, 476309310660, 7173360, 1

Offset: 0

Author

Geoffrey Critzer, Dec 02 2024

Keywords

• nonn
• tabl

Comments

A matrix M is idempotent if M^2 = M.

Examples

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,   12,      1;
  1,  117,    117,      1;
  1, 1080,  10530,   1080,    1;
  1, 9801, 882090, 882090, 9801, 1;
  ...
		

Crossrefs

Cf. A296548, A053846 (row sums).

Cf. A022167, A118180.

Programs

Maple

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
    end:
T:= (n,k)-> 3^(k*(n-k))*b(n, k):
seq(seq(T(n,k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024

Mathematica

nn = 8; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_, q_] := \[Gamma][n, q]/(q - 1)^n; \[Zeta][x_] := Sum[x^n/B[n, 3], {n, 0, nn}];Map[Select[#, # > 0 &] &, Table[B[n, 3], {n,0,nn}]*CoefficientList[Series[\[Zeta][x] \[Zeta][y x], {x, 0, nn}], {x, y}]] // Flatten

Formula

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*e(y*x) where e(x) = Sum_{n>=0} x^n/B(n) and B(n) = A053290(n)/2^n.

T(n,k) = A022167(n,k) * A118180(n,k). - Alois P. Heinz, Dec 02 2024