cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

1, 3, 12, 66, 513, 5769, 95706, 2379348, 89759799, 5188919427, 463209471288, 64236626341974, 13903296824817117, 4713694025825766861, 2510421030027019810854, 2104931848782489253483752, 2783505220978001187684672531, 5813031971452642599096778614183
Offset: 0

Views

Author

Geoffrey Critzer, Jul 12 2017

Keywords

Comments

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

Crossrefs

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

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

Formula

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

Views

Author

Geoffrey Critzer, Jul 14 2017

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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]

Formula

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

Views

Author

Geoffrey Critzer, Jul 07 2017

Keywords

Examples

			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;

Crossrefs

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

Programs

  • Mathematica
    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

Formula

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

Views

Author

Geoffrey Critzer, Jul 10 2017

Keywords

Comments

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...

Crossrefs

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

Programs

  • Mathematica
    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]]

Formula

Let A(x) = Sum_{n>=0} Product_{i>=n+1} (1-1/2^i)*x^n/A002884(n). Then A'(1) = 0.85017983...
Showing 1-4 of 4 results.

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