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A379778 Triangular array read by rows, T(n,k) is the number of n X n matrices over GF(2) that converge to an idempotent at rank k, n>=0, 0<=k<=n.

%I A379778 #18 Jul 12 2025 16:09:04
%S A379778 1,1,1,4,6,1,64,112,28,1,4096,7680,2240,120,1,1048576,2031616,634880,
%T A379778 39680,496,1,1073741824,2113929216,682622976,45711360,666624,2016,1,
%U A379778 4398046511104,8727373545472,2863669444608,198155698176,3096182784,10924032,8128,1
%N A379778 Triangular array read by rows, T(n,k) is the number of n X n matrices over GF(2) that converge to an idempotent at rank k, n>=0, 0<=k<=n.
%F A379778 Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = f(x)*e(y*x) where f(x) = Sum_{n>=0} q^(n^2-n)*x^n/B(n), e(x) = Sum_{n>=0} x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n, and q=2.
%e A379778 Triangle begins:
%e A379778         1;
%e A379778         1,       1;
%e A379778         4,       6,     1;
%e A379778        64,     112,     28,    1;
%e A379778      4096,    7680,   2240,   120,   1;
%e A379778   1048576, 2031616, 634880, 39680, 496, 1;
%e A379778   ...
%t A379778 nn = 7; q = 2; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_] := \[Gamma][n, q]/(q - 1)^n; e[u_] := Sum[u^n/B[n], {n, 0, nn}];
%t A379778 f[u_] := Sum[q^(n^2 - n) u^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &,
%t A379778   Table[B[n], {n, 0, nn}] CoefficientList[Series[f[u] e[ t u], {u, 0, nn}], {u, t}]] // Flatten
%Y A379778 Cf. A358649 (row sums), A053763 (column k=0).
%K A379778 nonn,tabl
%O A379778 0,4
%A A379778 _Geoffrey Critzer_, Jan 02 2025