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 1-7 of 7 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...

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

Views

Author

Geoffrey Critzer, Jul 15 2017

Keywords

Crossrefs

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

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

Formula

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.

A242938 Decimal expansion of c_e, coefficient associated with the asymptotic evaluation c_e*2^(n^2/4) of the number of subspaces of the n-dimensional vector space over the finite field F_2, n being even.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 27 2014

Keywords

Examples

			7.3719688014613165091531912...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.7 Lengyel's constant, p. 318.

Links

Crossrefs

Cf. A048651, A086053, A242939, A182176.

Programs

  • Mathematica
    digits = 100; EllipticTheta[3, 0, 1/2]/NProduct[1-2^(-j), {j, 1, Infinity}, WorkingPrecision -> digits + 10, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First
RealDigits[EllipticTheta[3, 0, 1/2]/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 17 2016 *)
  • PARI
    th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(1/2)/prodinf(n=1,1-2.^-n) \\ Charles R Greathouse IV, Jun 06 2016

Formula

(Sum_(k=-infinity..infinity) q^(-k^2)) / (prod_(j>0) (1-q^(-j))), with q = 2.

A242939 Decimal expansion of c_o, coefficient associated with the asymptotic evaluation c_o*2^(n^2/4) of the number of subspaces of the n-dimensional vector space over the finite field F_2, n being odd.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, May 27 2014

Keywords

Examples

			7.37194949076622733754141...

References

  • Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.7 Lengyel's constant, p. 318.

Links

Crossrefs

Cf. A048651, A086053, A242938, A182176.

Programs

  • Mathematica
    digits = 100; EllipticTheta[2, 0, 1/2]/NProduct[1 - 2^(-j), {j, 1, Infinity}, WorkingPrecision -> digits + 10, NProductFactors -> digits] // RealDigits[#, 10, digits]& // First
RealDigits[EllipticTheta[2, 0, 1/2]/QPochhammer[1/2, 1/2], 10, 100][[1]] (* Vladimir Reshetnikov, Oct 17 2016 *)
Showing 1-7 of 7 results.

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