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A298399 Triangle read by rows: T(n,k) is the number of direct sum decompositions of GF(2)^n whose maximal subspace has dimension k, 1<=k<=n, n>=1.

%I A298399 #19 Aug 02 2018 15:28:10
%S A298399 1,3,1,28,28,1,840,1960,120,1,83328,416640,39680,496,1,27998208,
%T A298399 295536640,40354560,666624,2016,1,32509919232,733279289344,
%U A298399 138360668160,2757537792,10924032,8128,1,132640470466560,6568159593103360,1654847774392320,38430207737856,181463777280,176865280,32640,1
%N A298399 Triangle read by rows: T(n,k) is the number of direct sum decompositions of GF(2)^n whose maximal subspace has dimension k, 1<=k<=n, n>=1.
%H A298399 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A298399 David Ellerman, <a href="http://arxiv.org/abs/1603.07619">The number of direct-sum decompositions of a finite vector space</a>, arXiv:1603.07619 [math.CO], 2016.
%e A298399 Triangle begins:
%e A298399       1;
%e A298399       3,      1;
%e A298399      28,     28,     1;
%e A298399     840,   1960,   120,   1;
%e A298399   83328, 416640, 39680, 496,  1;
%e A298399   ...
%t A298399 nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[ QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
%t A298399   Drop[Transpose[Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k + 1}]] -
%t A298399 Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}], z], {k, 0, 4}]], 1]]]
%Y A298399 Cf. A053601 (column 1), A270881 (row sums), A298561.
%K A298399 nonn,tabl
%O A298399 1,2
%A A298399 _Geoffrey Critzer_, Jan 18 2018