A303535 Triangle read by rows: T(n,k) is the number of ordered direct sum decompositions of the vector space GF(2)^n containing exactly k subspaces.

1, 0, 1, 0, 1, 6, 0, 1, 56, 168, 0, 1, 800, 10080, 20160, 0, 1, 20832, 1011840, 6666240, 9999360, 0, 1, 1051584, 216319488, 3839754240, 16798924800, 20158709760, 0, 1, 102233984, 106867464192, 5017364201472, 50390374809600, 163849992929280, 163849992929280

Offset: 0

Author

Geoffrey Critzer, Apr 25 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,     6;
  0, 1,    56,     168;
  0, 1,   800,   10080,   20160;
  0, 1, 20832, 1011840, 6666240, 9999360;
		

Crossrefs

Cf. A303533 (row sums).

Programs

Mathematica

nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[ QFactorial[n, q]] /. q -> 2; \[CapitalGamma][z_] := Sum[z^k/\[Gamma][k], {k, 0, nn}];Table[Take[(Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[1/(1 - u (\[CapitalGamma][z] - 1)), {z, 0, nn}], {z,u}])[[i]], i], {i, 1, nn + 1}] // Grid

Formula

Sum_{n>=0} T(n,k)y^k*x^n/g(n) = 1/(2-y(Sum_{n>=0} x^n/g(n)) where g(n) = A002884(n).