A289546 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.

1, 0, 1, 0, 1, 3, 0, 1, 14, 21, 0, 1, 65, 315, 315, 0, 1, 372, 4650, 13020, 9765, 0, 1, 2823, 87234, 527310, 1025325, 615195, 0, 1, 29210, 2291715, 27448764, 105413175, 156259530, 78129765, 0, 1, 417197, 88508205, 2043137265, 14019952275, 38897461575, 46487210175, 19923090075

Offset: 0

Author

Geoffrey Critzer, Jul 28 2017

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,    3;
  0, 1,   14,    21;
  0, 1,   65,   315,    315;
  0, 1,  372,  4650,  13020,    9765;
  0, 1, 2823, 87234, 527310, 1025325, 615195;
		

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A005329 (main diagonal), A289545 (row sums).

Programs

Mathematica

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

Formula

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.