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

0,4

Links

Table of n, a(n) for n=0..35.

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.

Example

       1;
       1,       1;
       4,       6,     1;
      64,     112,     28,    1;
    4096,    7680,   2240,   120,   1;
 1048576, 2031616, 634880, 39680, 496, 1;
 ...

Mathematica

nn = 7; q = 2; 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}]] // Grid

Crossrefs

Cf. A358649 (row sums), A053763 (column k=0).

Sequence in context: A193293 A258220 A158391 * A016491 A347052 A375366

Adjacent sequences: A379775 A379776 A379777 * A379779 A379780 A379781

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 02 2025

Status

approved