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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.

%I A378666 #22 Jul 12 2025 16:09:24
%S A378666 1,1,1,1,12,1,1,117,117,1,1,1080,10530,1080,1,1,9801,882090,882090,
%T A378666 9801,1,1,88452,72243171,666860040,72243171,88452,1,1,796797,
%U A378666 5873190687,491992666011,491992666011,5873190687,796797,1,1,7173360,476309310660,360089838858960,3267815287645062,360089838858960,476309310660,7173360,1
%N 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.
%C A378666 A matrix M is idempotent if M^2 = M.
%F A378666 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 A378666 T(n,k) = A022167(n,k) * A118180(n,k). - _Alois P. Heinz_, Dec 02 2024
%e A378666 Triangle T(n,k) begins:
%e A378666   1;
%e A378666   1,    1;
%e A378666   1,   12,      1;
%e A378666   1,  117,    117,      1;
%e A378666   1, 1080,  10530,   1080,    1;
%e A378666   1, 9801, 882090, 882090, 9801, 1;
%e A378666   ...
%p A378666 b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A378666       `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
%p A378666     end:
%p A378666 T:= (n,k)-> 3^(k*(n-k))*b(n, k):
%p A378666 seq(seq(T(n,k), k=0..n), n=0..8);  # _Alois P. Heinz_, Dec 02 2024
%t A378666 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
%Y A378666 Cf. A296548, A053846 (row sums).
%Y A378666 Cf. A022167, A118180.
%K A378666 nonn,tabl
%O A378666 0,5
%A A378666 _Geoffrey Critzer_, Dec 02 2024