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.

A053846 Number of n X n matrices over GF(3) of order dividing 2 (i.e., number of solutions of X^2=I in GL(n,3)).

Original entry on oeis.org

1, 2, 14, 236, 12692, 1783784, 811523288, 995733306992, 3988947598331024, 43581058503809001248, 1559669026899267564563936, 152805492791495918971070907584, 49094725258525117931062810300451648, 43237014297639482582550110281347475757696, 124920254287369111633119733942816364074145497472
Offset: 0

Views

Author

Vladeta Jovovic, Mar 28 2000

Keywords

Comments

Or, number of n X n invertible diagonalizable matrices over GF(3).

Examples

			a(2) = 14 because we have: {{0, 1}, {1, 0}}, {{0, 2}, {2, 0}}, {{1, 0}, {0, 1}}, {{1, 0}, {0,2}}, {{1, 0}, {1, 2}}, {{1, 0}, {2, 2}}, {{1, 1}, {0, 2}}, {{1,2}, {0, 2}}, {{2, 0}, {0, 1}}, {{2, 0}, {0, 2}}, {{2, 0}, {1,1}}, {{2, 0}, {2, 1}}, {{2, 1}, {0, 1}}, {{2, 2}, {0, 1}}. - _Geoffrey Critzer_, Aug 05 2017

References

  • Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Crossrefs

Cf. A053722, A053290, A290516.
Row sums of A378666.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, T(n-1, k-1)+3^k*T(n-1, k)))
    end:
a:= n-> add(3^(k*(n-k))*T(n, k), k=0...n):
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 06 2017
  • Mathematica
    nn = 14; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 3; G[z_] := Sum[z^k/g[k], {k, 0, nn}];Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* Geoffrey Critzer, Aug 05 2017 *)
  • PARI
    a(n)={my(v=[1]); for(n=1,n,v=vector(#v+1,k,if(k>1, v[k-1]) + if(k<=#v, 3^(k-1)*v[k]))); sum(k=0,n,3^(k*(n-k))*v[k+1])} \\ Andrew Howroyd, Mar 02 2018
  • Python
    from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if k<0 or k>n else 1 if n==0 else T(n - 1, k - 1) + 3**k*T(n - 1, k)
def a(n): return sum(3**(k*(n - k))*T(n, k) for k in range(n + 1))
print([a(n) for n in range(15)]) # Indranil Ghosh, Aug 06 2017, after Maple code

Formula

a(n)/A053290(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A053290(n))^2. - Geoffrey Critzer, Aug 05 2017

Extensions

More terms from Geoffrey Critzer, Aug 05 2017

A296605 Rectangle read by rows: T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have exactly k distinct eigenvalues, n >= 0, 0 <= k <= 3.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 36, 0, 0, 3, 702, 1404, 0, 3, 38070, 379080, 0, 3, 5351346, 341368830, 0, 3, 2434569858, 1231457092866, 0, 3, 2987199920970, 17481694843567584, 0, 3, 11966842794993066, 1077553466091961763220
Offset: 0

Views

Author

Geoffrey Critzer, Dec 16 2017

Keywords

Examples

			Array begins:
  1, 0,       0,         0,
  0, 3,       0,         0,
  0, 3,      36,         0,
  0, 3,     702,      1404,
  0, 3,   38070,    379080,
  0, 3, 5351346, 341368830

Links

Crossrefs

Cf. A290516 (row sums).

Programs

  • Mathematica
    nn = 8; g[ n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
    QFactorial[n, q]] /. q -> 3; G[u_, z_] := Sum[z^k/\[Gamma][k], {k, 0, nn}] - 1 + u ; Grid[Map[Reverse,Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[G[u, z]^3, {z, 0, nn}], {z, u}]]]

Formula

T(n,k)/A053290(n) is the coefficient of y^(3-k)*x^n in the expansion of (-1 + y + Sum_{n>=0} x^n/A053290(n))^3.

A297892 Triangle read by rows. T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have rank k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 1, 2, 1, 24, 14, 1, 234, 1638, 236, 1, 2160, 147420, 254880, 12692, 1, 19602, 12349260, 208173240, 124394292, 1783784, 1, 176904, 1011404394, 157378969440, 916910326332, 157779262368, 811523288
Offset: 0

Views

Author

Geoffrey Critzer, Jan 07 2018

Keywords

Examples

			Triangle begins
  1;
  1,     2;
  1,    24,       14;
  1,   234,     1638,       236;
  1,  2160,   147420,    254880,     12692;
  1, 19602, 12349260, 208173240, 124394292, 1783784;

Links

Crossrefs

Cf. A296548, A053846 (main diagonal), A290516 (row sums).

Programs

  • Mathematica
    nn = 5; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 3;G[n] := Sum[u z^r/g[r], {r, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}]^2 Sum[
       z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]

Formula

T(n,k)/A053290(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A053290(n) * (Sum_{n>=0} y*x^n\A053290(n))^2.

A346421 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(3) such that the sum of the dimensions of its eigenspaces taken over all its eigenvalues is k, 0 <= k <= n, n >= 0.

Original entry on oeis.org

1, 0, 3, 18, 24, 39, 3456, 8190, 5928, 2109, 7619508, 17094240, 13700700, 4215120, 417153, 149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179, 26394940582090344, 59388527912287392, 47325384827973252, 15262273318168800, 1648005959253654, 74268805562952, 1233891662727
Offset: 0

Views

Author

Geoffrey Critzer, Jul 16 2021

Keywords

Examples

			             1;
             0,            3;
            18,           24,           39;
          3456,         8190,         5928,        2109;
       7619508,     17094240,     13700700,     4215120,     417153;
  149200289280, 335730157884, 267485755680, 85615372260, 8910314160, 346720179;

Links

Crossrefs

Cf. A346209 (column k=0), A290516 (main diagonal), A060722 (row sums).

Programs

  • Mathematica
    nn = 7; q = 3; b[p_, i_] := Count[p, i]; d[p_, i_] :=Sum[j b[p, j], {j, 1, i}] + i Sum[b[p, j], {j, i + 1, Total[p]}];aut[deg_, p_] := Product[Product[q^(d[p, i] deg) - q^((d[p, i] - k) deg), {k, 1, b[p, i]}], {i, 1,Total[p]}]; A001037 =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; g[u_, v_] :=
Total[Map[v^Length[#] u^Total[#]/aut[1, #] &, Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]]]; Table[Take[(Table[ Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[g[u, v]^3 Product[Product[1/(1 - (u/q^r)^d), {r, 1, \[Infinity]}]^A001037[[d]], {d, 2, nn}], {u, 0, nn}], {u, v}])[[n]],n], {n, 1, nn}] // Grid
Showing 1-4 of 4 results.

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