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A298561 Triangle read by rows. T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.

%I A298561 #17 Aug 02 2018 15:24:48
%S A298561 1,3,4,28,56,57,840,2800,2920,2921,83328,499968,539648,540144,540145,
%T A298561 27998208,323534848,363889408,364556032,364558048,364558049,
%U A298561 32509919232,765789208576,904149876736,906907414528,906918338560,906918346688,906918346689
%N A298561 Triangle read by rows.  T(n,k) is the number of direct sum decompositions of GF(2)^n into subspaces of dimension at most k, 1<=k<=n.
%H A298561 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 A298561 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.
%F A298561 exp(Sum_{j=0...k} x^j/A002884(j)) = Sum_{n>=0} T(n,k)/A002884(n)*x^n.
%e A298561   1
%e A298561   3,     4,
%e A298561   28,    56,     57,
%e A298561   840,   2800,   2920,   2921,
%e A298561   83328, 499968, 539648, 540144, 540145,
%t A298561 nn = 7; \[Gamma][n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 2; Flatten[Table[Table[Transpose[
%t A298561      Map[Drop[#, 1] &,Table[Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[Exp[Sum[z^i/\[Gamma][i], {i, 1, k}]], {z, 0, nn}],z], {k, 1, nn}]]][[j, k]], {k, 1, j}], {j, 1, nn}]]
%Y A298561 Cf. A270881 (main diagonal), A053601 (column 1), A298339.
%K A298561 nonn,tabl
%O A298561 1,2
%A A298561 _Geoffrey Critzer_, Jan 21 2018