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%I #16 Jul 30 2017 03:40:52
%S 1,0,1,0,1,3,0,1,14,21,0,1,65,315,315,0,1,372,4650,13020,9765,0,1,
%T 2823,87234,527310,1025325,615195,0,1,29210,2291715,27448764,
%U 105413175,156259530,78129765,0,1,417197,88508205,2043137265,14019952275,38897461575,46487210175,19923090075
%N Triangle read by rows. T(n,k) is the number of flags in an n dimensional vector space over GF(2) that have length exactly k, n >= 0, 0 <= k <= n.
%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 T(n,k)/A005329(n) is the coefficient of y^k*x^n in 1/(1 - y (eq(x) - 1)) where eq(x) is the q-exponential function.
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 3;
%e 0, 1, 14, 21;
%e 0, 1, 65, 315, 315;
%e 0, 1, 372, 4650, 13020, 9765;
%e 0, 1, 2823, 87234, 527310, 1025325, 615195;
%t nn = 8; eq[z_] := Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[Take[(Table[ FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[ 1/(1 - u (eq[z] - 1)) /. q -> 2, {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
%Y Cf. A005329 (main diagonal), A289545 (row sums).
%K nonn,tabl
%O 0,6
%A _Geoffrey Critzer_, Jul 28 2017
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