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A289545 Number of flags in an n-dimensional vector space over GF(2).

%I A289545 #29 Jun 25 2025 22:09:03
%S A289545 1,1,4,36,696,27808,2257888,369572160,121459776768,79991977040128,
%T A289545 105466641591287296,278244130564826548224,1468496684404408240109568,
%U A289545 15502543140842029367582248960,327332729703063815298568073396224,13823536566775628445052117519260598272
%N A289545 Number of flags in an n-dimensional vector space over GF(2).
%H A289545 Vincenzo Librandi, <a href="/A289545/b289545.txt">Table of n, a(n) for n = 0..80</a>
%H A289545 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 A289545 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 A289545 a(n) = Sum A005329(n)/( A005329(n_1)*A005329(n_2)*...*A005329(n_k) ) where the sum is over all compositions of n = n_1 + n_2 + ... + n_k.
%F A289545 G.f. a(n)/A005329(n) is the coefficient of x^n in 1/(2 - eq(x)) where eq(x) is the 2-exponential function.
%F A289545 a(n) = Sum_{k=0..binomial(n,2)} 2^k * A381299(n,k). - _Alois P. Heinz_, Feb 21 2025
%p A289545 b:= proc(o, u, t) option remember; `if`(u+o=0, 1, `if`(t=1,
%p A289545       b(u+o, 0$2), 0)+add(2^(u+j-1)*b(o-j, u+j-1, 1), j=1..o))
%p A289545     end:
%p A289545 a:= n-> b(n, 0$2):
%p A289545 seq(a(n), n=0..16);  # _Alois P. Heinz_, Feb 21 2025
%t A289545 nn = 15; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0,
%t A289545    nn}] CoefficientList[Series[ 1/(1 - (eq[z] - 1)) /. q -> 2, {z, 0, nn}], z]
%Y A289545 Cf. A381299.
%Y A289545 Column k=2 of A381426.
%K A289545 nonn
%O A289545 0,3
%A A289545 _Geoffrey Critzer_, Jul 28 2017