A289538 -id:A289538 - cp's OEIS frontend

A289539 Number of ways to choose a subspace U of GF(2)^n and then choose a subspace of U.

1, 3, 12, 66, 513, 5769, 95706, 2379348, 89759799, 5188919427, 463209471288, 64236626341974, 13903296824817117, 4713694025825766861, 2510421030027019810854, 2104931848782489253483752, 2783505220978001187684672531, 5813031971452642599096778614183

Offset: 0

Geoffrey Critzer, Jul 12 2017

nonn

A q-analog (q=2) of A000244.

Cf. A000244, A182176, A289537, A289538, A289541, A289542.

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]^3 /. q -> 2, {z, 0, nn}], z]

a(n) = Sum_{k=0..n} A022166(n,k)*A006116(k).

a(n)/[n]_q! is the coefficient of x^n in the expansion of exp_q(x)^3 when q -> 2 and where exp_q(x) is the q-exponential function and [n]_q! is the q-factorial of n.

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

Original entry on oeis.org

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).

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.

cp's OEIS Frontend

A289539 Number of ways to choose a subspace U of GF(2)^n and then choose a subspace of U.

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

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