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.

A346164 Number of n X n matrices over GF(2) whose characteristic polynomial is squarefree.

Original entry on oeis.org

1, 2, 8, 160, 22272, 9744384, 20309999616, 165823024988160, 5334245506774204416, 699753231745207240753152, 366124801432291852761377538048, 769585907704777340287352115528990720, 6438115769123814066544745845515649290338304, 216154104085428332447218371078526172108859761491968
Offset: 0

Views

Author

Geoffrey Critzer, Jul 08 2021

Keywords

Links

Crossrefs

Cf. A001037, A002884, A346082.

Programs

  • Mathematica
    nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}]; Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[Product[(1 + x^i/(2^i - 1))^A001037[[i]], {i, 1, nn}], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n)*x^n/A002884(n) = Product_{n>=1} (1 + x^n/(2^n-1))^A001037(n).
Lim_{n->infinity} a(n)/A002884(n) = 1. - Geoffrey Critzer, Oct 21 2021

A346165 Number of n X n invertible matrices whose characteristic polynomial is squarefree.

Original entry on oeis.org

1, 1, 2, 104, 9792, 4887552, 10456694784, 80831009783808, 2695921347430711296, 347083584759711311855616, 184330749741189300682890412032, 383205061911277693825526401937178624, 3224343525101169010615339144085384529444864, 107976295438859678148286176040509108456782680817664
Offset: 0

Views

Author

Geoffrey Critzer, Jul 08 2021

Keywords

Links

Crossrefs

Cf. A001037, A002884, A346082, A346164.

Programs

  • Mathematica
    nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[(1 + x) Product[(1 + x^i/(2^i - 1))^A001037[[i]], {i, 2,nn}], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n)*x^n/A002884(n) = (1 + x) * Product_{n>=2} (1 + x^n/(2^n -1))^A001037(n)
Lim_{n->infinity} a(n)/A002884(n) = 1/2. - Geoffrey Critzer, Oct 21 2021

A347010 Triangular array read by rows. T(n,k) is the number of n X n matrices over GF(2) with minimal polynomial of degree k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 2, 14, 0, 2, 98, 412, 0, 2, 1542, 13160, 50832, 0, 2, 34782, 1147744, 6854720, 25517184, 0, 2, 1908734, 260411904, 2544075264, 14153094144, 51759986688, 0, 2, 166738046, 107691724672, 2985421682688, 21570911944704, 116285097148416, 422000664182784
Offset: 0

Views

Author

Geoffrey Critzer, Aug 10 2021

Keywords

Examples

			  1,
  0, 2,
  0, 2,   14,
  0, 2,   98,   412,
  0, 2, 1542, 13160, 50832

Links

Crossrefs

Cf. A002416 (row sums), A346082 (main diagonal).

Programs

  • Mathematica
    nn = 8; q = 2; 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]}]; \[Nu] =
Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}]; L = Level[Table[IntegerPartitions[n], {n, 0, nn}], {2}]; g[u_, v_, deg_] :=
Total[Map[v^(Max[Prepend[#, 0]] deg) u^(deg Total[#])/aut[deg, #] &,
   L]]; Table[Take[(Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[g[u, v, deg]^\[Nu][[deg]], {deg, 1, nn}], {u, 0, nn}], {u, v}])[[n]], n], {n, 1, nn}] // Grid

A346084 Number of invertible n X n cyclic matrices over GF(2).

Original entry on oeis.org

1, 1, 5, 146, 17352, 8607552, 17362252800, 141087882903552, 4605333486987902976, 602440395156024780128256, 315546297657431573076891402240, 661423879140352987222707528171257856, 5547073628722488310034844542685201882415104, 186106159461598495645613441708238958650047305089024
Offset: 0

Views

Author

Geoffrey Critzer, Jul 04 2021

Keywords

Links

Crossrefs

Cf. A001037, A002884, A346082.

Programs

  • Mathematica
    nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}]CoefficientList[Series[(1 + 2 x/(2 - x)) Product[(1 + 2^i x^i/((2^i - 1) (2^i - x^i)))^ A001037[[i]], {i, 2, nn}], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n)x^n/A002884(n) = (1 + 2x/(2-x)* Product_{i>=2}(1 + x^i/((2^i-1)(1-x/2)^i)))^A001037(i).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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