A287206 - OEIS

%I #16 Aug 02 2017 11:34:13

%S 1,0,1,1,0,3,1,28,0,28,281,120,1680,0,840,9921,139376,29760,277760,0,

%T 83328,16078337,20000736,140491008,19998720,139991040,0,27998208,

%U 13596908545,130684723136,81282991104,380636971008,40637399040,227569434624,0,32509919232,191426147495937,443803094908800,2132774681579520,884358943211520,3105997683425280,265280940933120,1237977724354560,0,132640470466560

%N 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 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 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 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(n) is A002884(n).

%e   1,

%e   0,        1,

%e   1,        0,        3,

%e   1,        28,       0,         28,

%e   281,      120,      1680,      0,       840,

%e   9921,     139376,   29760,    277760,   0,    83328,

%t 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 Table[Table[(Table[g[n], {n, 0, nn}] CoefficientList[  Series[Exp[e[u] - 1 - u + u t], {u, 0, nn}], {u, t}])[[n,

%t      k]], {k, 1, n}], {n, 1, nn + 1}] // Grid

%Y Cf. A270881 (row sums), A053601 (main diagonal), A289543, A289540 (column k=0).

%K nonn,tabl

%O 0,6

%A _Geoffrey Critzer_, May 21 2017