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A287206 Triangle read by rows: T(n,k) is the number of direct sum decompositions of a finite vector space of n dimensions over GF(2) that have exactly k subspaces of dimension 1, n>=0, 0<=k<=n.

%I A287206 #22 Jan 15 2025 14:48:28
%S A287206 1,0,1,1,0,3,1,28,0,28,281,120,1680,0,840,9921,139376,29760,277760,0,
%T A287206 83328,16078337,20000736,140491008,19998720,139991040,0,27998208,
%U A287206 13596908545,130684723136,81282991104,380636971008,40637399040,227569434624,0,32509919232,191426147495937,443803094908800,2132774681579520,884358943211520,3105997683425280,265280940933120,1237977724354560,0,132640470466560
%N A287206 Triangle read by rows: T(n,k) is the number of direct sum decompositions of a finite vector space of n dimensions over GF(2) that have exactly k subspaces of dimension 1, n>=0, 0<=k<=n.
%H A287206 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.
%H A287206 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1
%F A287206 Sum_{n>=0} T(n,k)*u^n/g(n)*t^k = exp(Sum_{r>=0} u^r/g(r) - 1 - u + t*u) where g = A002884.
%e A287206 Triangle T(n,k) begins:
%e A287206      1;
%e A287206      0,      1;
%e A287206      1,      0,     3;
%e A287206      1,     28,     0,     28,
%e A287206    281,    120,  1680,      0,  840;
%e A287206   9921, 139376, 29760, 277760,    0, 83328;
%e A287206   ...
%t A287206 nn = 8; g[n_] := QFactorial[n, q]*(q - 1)^n*q^Binomial[n, 2] /. q -> 2; e[u_] := Sum[u^r/g[r], {r, 0, nn}];
%t A287206 Table[Table[(Table[g[n], {n, 0, nn}] CoefficientList[  Series[Exp[e[u] - 1 - u + u t], {u, 0, nn}], {u, t}])[[n,
%t A287206      k]], {k, 1, n}], {n, 1, nn + 1}] // Grid
%Y A287206 Cf. A270881 (row sums), A053601 (main diagonal), A289543 (column k=0).
%K A287206 nonn,tabl
%O A287206 0,6
%A A287206 _Geoffrey Critzer_, May 21 2017