A378666 - OEIS

%I #16 Dec 03 2024 08:42:41

%S 1,1,1,1,12,1,1,117,117,1,1,1080,10530,1080,1,1,9801,882090,882090,

%T 9801,1,1,88452,72243171,666860040,72243171,88452,1,1,796797,

%U 5873190687,491992666011,491992666011,5873190687,796797,1,1,7173360,476309310660,360089838858960,3267815287645062,360089838858960,476309310660,7173360,1

%N 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.

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

%F 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.

%F T(n,k) = A022167(n,k) * A118180(n,k). - _Alois P. Heinz_, Dec 02 2024

%e Triangle T(n,k) begins:

%e   1;

%e   1,    1;

%e   1,   12,      1;

%e   1,  117,    117,      1;

%e   1, 1080,  10530,   1080,    1;

%e   1, 9801, 882090, 882090, 9801, 1;

%e   ...

%p b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,

%p       `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))

%p     end:

%p T:= (n,k)-> 3^(k*(n-k))*b(n, k):

%p seq(seq(T(n,k), k=0..n), n=0..8);  # _Alois P. Heinz_, Dec 02 2024

%t nn = 7; 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}]] // Grid

%Y Cf. A296548, A053846 (row sums).

%Y Cf. A022167, A118180.

%K nonn,tabl,new

%O 0,5

%A _Geoffrey Critzer_, Dec 02 2024