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.
1, 1, 1, 4, 6, 1, 64, 112, 28, 1, 4096, 7680, 2240, 120, 1, 1048576, 2031616, 634880, 39680, 496, 1, 1073741824, 2113929216, 682622976, 45711360, 666624, 2016, 1, 4398046511104, 8727373545472, 2863669444608, 198155698176, 3096182784, 10924032, 8128, 1
Offset: 0
Examples
Triangle begins:
1;
1, 1;
4, 6, 1;
64, 112, 28, 1;
4096, 7680, 2240, 120, 1;
1048576, 2031616, 634880, 39680, 496, 1;
...
Programs
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Mathematica
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}]; f[u_] := Sum[q^(n^2 - n) u^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[f[u] e[ t u], {u, 0, nn}], {u, t}]] // Flatten
Formula
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.