A289383 Total number of nonzero vectors over all subspaces of an n-dimensional vector space over the field with two elements.

0, 1, 6, 35, 240, 2077, 23562, 358775, 7449060, 213188689, 8473977534, 470309723435, 36582636406680, 3998655357260293, 615328930033081458, 133485330929459963615, 40859530900982506959180, 17659495180812130332490681, 10781678259164073608877557286, 9301770545157096607562560360595

Offset: 0

Author

Geoffrey Critzer, Jul 04 2017

Keywords

• nonn

Comments

The q-analog of A001787.

Crossrefs

Cf. A001787, A006116, A022166.

Programs

Mathematica

nn = 20; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[z eq[z]^2 /. q -> 2, {z, 0, nn}], z]

Formula

a(n) = Sum_{k=1..n} A022166(n,k)*(2^k - 1).

a(n)/[n]_q! is the coefficient of x^n in the expansion of x*exp_q(x)^2 when q->2 and where exp_q(x) is the q exponential function and [n]_q! is the q-factorial of n.

a(n) = (2^n - 1)*A006116(n-1).