A289539 -id:A289539 - cp's OEIS frontend

A289541 Number of subspaces of GF(2)^n with even dimension.

1, 1, 2, 8, 37, 187, 1304, 14606, 222379, 4141729, 107836478, 4466744372, 258501941713, 18779494904263, 1918824942497636, 311738238353418074, 71234670515346760951, 20564497734374127115501, 8363824677163863282113162, 5408580882753786431279731328

Offset: 0

Geoffrey Critzer, Jul 14 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A182176, A289537, A289538, A289539, A289542.

nn = 22; eq[z_] := Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];
coshq[z_] := Sum[z^(2 n)/FunctionExpand[QFactorial[(2 n), q]], {n, 0, nn}];
Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}]*
CoefficientList[Series[coshq[z]*eq[z] /. q -> 2, {z, 0, nn}], z]

a(n)/[n]q! is the coefficient of x^n in the expansion of exp_q(x)*cosh_q(x) when q->2, and cosh_q(x) = Sum{n>=0} x^(2n)/[2n]_q!, and exp_q(x) is the q-exponential function, and [n]_q! is the q-factorial of n.
From Vaclav Kotesovec, Jun 11 2025: (Start)
a(n) ~ c * 2^(n^2/4), where
c = 3.89569562162... if mod(n,4) = 0,
c = 3.68597474538... if mod(n,4) = 1 or 3,
c = 3.47627317983... if mod(n,4) = 2. (End)

A289537 Triangle read by rows: T(n,k) is the number of k-dimensional subspaces of an n-dimensional vector space over F_2 that do not contain a given nonzero vector, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 14, 28, 8, 0, 1, 30, 140, 120, 16, 0, 1, 62, 620, 1240, 496, 32, 0, 1, 126, 2604, 11160, 10416, 2016, 64, 0, 1, 254, 10668, 94488, 188976, 85344, 8128, 128, 0, 1, 510, 43180, 777240, 3212592, 3108960, 690880, 32640, 256, 0

Offset: 0

Geoffrey Critzer, Jul 07 2017

			Triangle begins:
  1;
  1,    0;
  1,    2,    0;
  1,    6,    4,    0;
  1,   14,   28,    8,    0;
  1,   30,  140,  120,   16,    0;
  1,   62,  620, 1240,  496,   32,    0;

Cf. A022166, A182176, A289538, A289539, A289541, A289542.

Table[Table[Product[q^n - q^i, {i, 1, k}]/Product[q^k - q^i, {i, 0, k - 1}] /. q -> 2, {k, 0, n}], {n, 0, 9}] // Grid

T(n,k) = 2^k * A022166(n-1,k).

A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.

Original entry on oeis.org

8, 5, 0, 1, 7, 9, 8, 3, 0, 8, 7, 3, 9, 7, 9, 3, 3, 2, 8, 7, 6, 0, 6, 3, 2, 8, 1, 4, 9, 3, 5, 9, 1, 8, 7, 8, 8, 4, 0, 4, 2, 6, 7, 2, 5, 9, 7, 3, 2, 0, 2, 7, 2, 5, 9, 8, 7, 3, 5, 8, 0, 5, 2, 5, 5, 6, 3, 0, 9, 5, 9, 4, 1, 1, 8, 3, 3, 1, 3, 4, 4, 3, 6, 3, 0, 4, 1, 0, 6, 7, 0, 8, 8, 5, 9, 3, 5, 6, 5, 8

Offset: 0

Geoffrey Critzer, Jul 10 2017

More precisely, let X:L(V) -> {0,1,2,...,n} be the random variable that assigns to each linear operator T on n-dimensional vector space V over F_2, the integer j in {0,1,2,...,n} such that the dimension of the null space of T = j. Then E(X) = 0.850179183...

Cf. A002884, A182176, A289537, A289539, A289541, A289542.

nn = 300; q := 2;A[x_] := Sum[1/(FunctionExpand[QFactorial[j, q]] (q - 1)^j q^Binomial[j, 2]) Product[1 - 1/q^i, {i, j + 1, \[Infinity]}] x^j, {j, 0, nn}];RealDigits[
  N[Normal[Series[D[A[x], x] /. x -> 1, {x, 0, nn}]], 100]][[1]]

Let A(x) = Sum_{n>=0} Product_{i>=n+1} (1-1/2^i)*x^n/A002884(n). Then A'(1) = 0.85017983...

A289542 Number of ordered pairs of nonzero vectors over the subspaces of GF(2)^n.

Original entry on oeis.org

0, 1, 12, 119, 1290, 16957, 285264, 6343523, 190424310, 7826128009, 444658035228, 35162773747631, 3888419271339330, 603295404971492053, 131635270366023841896, 40458451431717420232187, 17536781855825299937977230, 10728658644626168469625854241

Offset: 0

Geoffrey Critzer, Jul 15 2017

nonn

Cf. A182176, A289537, A289538, A289539, A289541.

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[ eq[z]^2 (z + 2 z^2) /. q -> 2, {z, 0, nn}], z]

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

A382223 Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u=0, k>=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 16, 12, 4, 1, 0, 1, 67, 66, 22, 5, 1, 0, 1, 374, 513, 172, 35, 6, 1, 0, 1, 2825, 5769, 1969, 355, 51, 7, 1, 0, 1, 29212, 95706, 33856, 5380, 636, 70, 8, 1, 0, 1, 417199, 2379348, 893188, 125090, 12006, 1036, 92, 9, 1

Offset: 0

Geoffrey Critzer, Mar 23 2025

			 1, 1,   1,    1,     1,      1,      1,...
 0, 1,   2,    3,     4,      5,      6,...
 0, 1,   5,   12,    22,     35,     51,...
 0, 1,  16,   66,   172,    355,    636,...
 0, 1,  67,  513,  1969,   5380,  12006,...
 0, 1, 374, 5769, 33856, 125090, 352476,...

Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, Discrete Mathematics, Vol. 343, No. 11 (2020), 112041; arXiv preprint, arXiv:1909.01550 [math.CO], 2019-2020. See Table 2.
R. P. Stanley, Acyclic orientation of graphs, Discrete Math. 5 (1973), 171-178. North Holland Publishing Company.

Cf. A006116 column k=2, A289539 column k=3, A005329, A382363.

nn = 6; B[n_] := QFactorial[n, 2];e[z_] := Sum[z^n/B[n], {n, 0, nn}]; zetapolys = Drop[Map[Expand[InterpolatingPolynomial[#, x]] &,Transpose[Table[Table[B[n], {n, 0, nn}] CoefficientList[Series[e[z]^k, {z, 0, nn}], z], {k, 1, nn}]]], -1];Table[zetapolys /. x -> i, {i, 0, nn}] // Transpose // Grid

Sum_{n>=0} T(n,k)/A005329(n) = e(x)^k, where e(x) = Sum_{n>=0}x^n/A005329(n).

cp's OEIS Frontend

A289541 Number of subspaces of GF(2)^n with even dimension.

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A289537 Triangle read by rows: T(n,k) is the number of k-dimensional subspaces of an n-dimensional vector space over F_2 that do not contain a given nonzero vector, n>=0, 0<=k<=n.

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A289538 Expected dimension of the null space of a random linear operator on an n-dimensional vector space over the field with two elements as n -> infinity.

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A289542 Number of ordered pairs of nonzero vectors over the subspaces of GF(2)^n.

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A382223 Rectangular array read by antidiagonals: T(n,k) is the number of labeled digraphs on [n] along with a (coloring) function c:[n] -> [k] with the property that for all u,v in [n], u->v implies u=0, k>=0.

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