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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.

%I A303535 #14 May 07 2018 03:53:04
%S A303535 1,0,1,0,1,6,0,1,56,168,0,1,800,10080,20160,0,1,20832,1011840,6666240,
%T A303535 9999360,0,1,1051584,216319488,3839754240,16798924800,20158709760,0,1,
%U A303535 102233984,106867464192,5017364201472,50390374809600,163849992929280,163849992929280
%N 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.
%F A303535 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).
%e A303535 Triangle begins:
%e A303535   1;
%e A303535   0, 1;
%e A303535   0, 1,     6;
%e A303535   0, 1,    56,     168;
%e A303535   0, 1,   800,   10080,   20160;
%e A303535   0, 1, 20832, 1011840, 6666240, 9999360;
%t A303535 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
%Y A303535 Cf. A303533 (row sums).
%K A303535 nonn,tabl
%O A303535 0,6
%A A303535 _Geoffrey Critzer_, Apr 25 2018