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.

A281773 Number of distinct topologies on an n-set that have exactly 4 open sets.

Original entry on oeis.org

0, 0, 1, 9, 43, 165, 571, 1869, 5923, 18405, 56491, 172029, 521203, 1573845, 4742011, 14266989, 42882883, 128812485, 386765131, 1160950749, 3484162963, 10455110325, 31370573851, 94122207309, 282387593443, 847204723365, 2541698056171, 7625261940669
Offset: 0

Views

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

Examples

			a(3) = 9 because we have: {{}, {c}, {a,b}, {a,b,c}} with 3 labelings and {{}, {c}, {b,c}, {a,b,c}} with 6 labelings.

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.
Partial sums are given in A298564.

Programs

  • Mathematica
    CoefficientList[Series[x^2*(1 + 3 x)/((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 27}], x] (* Michael De Vlieger, Jan 21 2018 *)
  • PARI
    a(n) = stirling(n,2,2) + 3!*stirling(n,3,2) \\ Colin Barker, Jan 30 2017
  • PARI
    concat(vector(2), Vec(x^2*(1 + 3*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = A000392(n+1) + 3*A000392(n).
E.g.f.: (exp(x)-1)^3 + (exp(x)-1)^2/2!.
From Colin Barker, Jan 30 2017: (Start)
G.f.: x^2*(1 + 3*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>3.
a(n) = 2 - 5*2^(n-1) + 3^n for n>0. (End)

A281774 Number of distinct topologies on an n-set with exactly 6 open sets.

Original entry on oeis.org

0, 0, 0, 6, 72, 630, 4680, 31206, 193032, 1131990, 6386760, 35025606, 188061192, 993760950, 5187840840, 26831095206, 137770476552, 703455087510, 3576115150920, 18117222864006, 91536570671112, 461496288791670, 2322770028381000, 11675109032796006
Offset: 0

Views

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Programs

  • Mathematica
    LinearRecurrence[{15,-85,225,-274,120},{0,0,0,6,72,630},30] (* Harvey P. Dale, Oct 22 2018 *)
  • PARI
    a(n) = 3!*stirling(n, 3, 2) + 3*4!*stirling(n, 4, 2)/2 + 5!*stirling(n, 5, 2) \\ Colin Barker, Jan 30 2017
  • PARI
    concat(vector(3), Vec(6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 3! Stirling2(n, 3) + 3/2*4! Stirling2(n, 4) + 5! Stirling2(n, 5).
From Colin Barker, Jan 30 2017: (Start)
a(n) = 2 - 2^(2+n) - 7*2^(2*n-1) + 5*3^n + 5^n for n>5.
a(n) = 15*a(n-1) - 85*a(n-2) + 225*a(n-3) - 274*a(n-4) + 120*a(n-5) for n>5.
G.f.: 6*x^3*(1 - 3*x + 10*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)).
(End)

A281775 Number of distinct topologies on an n-set that have exactly 7 open sets.

Original entry on oeis.org

0, 0, 0, 0, 54, 780, 7830, 67620, 535374, 3992940, 28483110, 196316340, 1317106494, 8650141500, 55853351190, 355770438660, 2241509994414, 13998294536460, 86795899256070, 535048203626580, 3282628800655134, 20061393719417820, 122212221633141750
Offset: 0

Views

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A281774, A028244, A281775, A281776, A281777, A281778, A281779, A281780.

Programs

  • PARI
    a(n) = 9*4!*stirling(n, 4, 2)/4 + 2*5!*stirling(n, 5, 2) + 6!*stirling(n, 6, 2) \\ Colin Barker, Jan 30 2017
  • PARI
    concat(vector(4), Vec(6*x^4*(9 - 59*x + 150*x^2) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 9/4*4! Stirling2(n, 4) + 2*5! Stirling2(n, 5) + 6! Stirling2(n, 6).
From Colin Barker, Jan 30 2017: (Start)
G.f.: 6*x^4*(9 - 59*x + 150*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)).
a(n) = 21*a(n-1) - 175*a(n-2) + 735*a(n-3) - 1624*a(n-4) + 1764*a(n-5) - 720*a(n-6) for n>6.
a(n) = -5 + 17*2^(n-1) - 3^(2+n) + 29*4^(n-1) - 4*5^n + 6^n for n>0. (End)

A281776 Number of distinct topologies on an n-set that have exactly 8 open sets.

Original entry on oeis.org

0, 0, 0, 1, 54, 955, 11760, 122941, 1175034, 10595215, 91506420, 763624081, 6194818014, 49084747075, 381338401080, 2914184784421, 21965095364994, 163656285828535, 1207613518375740, 8838842878371961, 64253768864671974, 464416229729871595, 3340518964319750400
Offset: 0

Views

Author

Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Programs

  • PARI
    concat(vector(3), Vec(x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = Stirling2(n, 3) + 2*4! Stirling2(n, 4) + 15/4*5! Stirling2(n, 5) + 5/2*6! Stirling2(n, 6) + 7! Stirling2(n, 7).
From Colin Barker, Jan 30 2017: (Start)
a(n) = 13/4 - 19*2^(n-1) + 44*3^(n-1) - 2^(n-1)*3^(2+n) - 57*4^(n-1) + (39*5^n)/4 + 7^n for n>0.
G.f.: x^3*(1 + 26*x - 235*x^2 + 448*x^3 + 2100*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)).
(End)

A281777 Number of distinct topologies on an n-set that have exactly 9 open sets.

Original entry on oeis.org

0, 0, 0, 0, 20, 800, 14260, 189280, 2181060, 23241120, 235737620, 2308206560, 21979728100, 204477713440, 1864504348980, 16707856095840, 147469451067140, 1284607771225760, 11063319237792340, 94343562846289120, 797685042851814180, 6694943490279586080
Offset: 0

Views

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

Programs

  • Mathematica
    LinearRecurrence[{36,-546,4536,-22449,67284,-118124,109584,-40320},{0,0,0,0,20,800,14260,189280,2181060},30] (* Harvey P. Dale, Aug 19 2020 *)
  • PARI
    concat(vector(4), Vec(20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 5/6*4! Stirling2(n, 4) + 5*5! Stirling2(n, 5) + 11/2*6! Stirling2(n, 6) + 3*7! Stirling2(n, 7) + 8! Stirling2(n, 8).
G.f.: 20*x^4*(1 + 4*x - 181*x^2 + 1100*x^3 - 1344*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)). - Colin Barker, Jan 30 2017

A281778 Number of distinct topologies on an n-set that have exactly 10 open sets.

Original entry on oeis.org

0, 0, 0, 0, 24, 900, 18030, 276570, 3680964, 45065160, 523292010, 5859909990, 63862084704, 680829769620, 7122705252390, 73284607133010, 742843170653244, 7429450873589280, 73416173732059170, 717721593866613630, 6949589106333898584, 66721599431782204140
Offset: 0

Views

Author

Submitted on behalf of Moussa Benoumhani by Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777,A281778, A281779, A281780.

Programs

  • PARI
    concat(vector(4), Vec(6*x^4*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*x^5) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)*(1 - 8*x)*(1 - 9*x)) + O(x^30))) \\ Colin Barker, Jan 30 2017

Formula

a(n) = 4! Stirling2(n, 4) + 11/2*5! Stirling2(n, 5) + 73/8*6! Stirling2(n, 6) + 15/2*7! Stirling2(n, 7) + 7/2*8! Stirling2(n, 8) + 9! Stirling2(n, 9).
G.f.: (6*(4 - 30*x - 265*x^2 + 3570*x^3 - 10839*x^4 + 22680*x^5))*x^4/Product_{j=1..9} (1-j*x). - Robert Israel, Jan 29 2017

A281780 Number of distinct topologies on an n-set that have exactly 12 open sets.

Original entry on oeis.org

0, 0, 0, 0, 12, 660, 20400, 445620, 7977732, 126860580, 1873839000, 26381789940, 359484471852, 4784481401700, 62538498859200, 805447464281460, 10241415118476372, 128722997969290020, 1600670708273985000, 19705915838479512180, 240330009637668935292
Offset: 0

Views

Author

Geoffrey Critzer, Jan 29 2017

Keywords

Links

Crossrefs

The number of distinct topologies on an n-set with exactly k open sets for k=2..12 is given by A000012, A000918, A281773, A028244, A281774, A281775, A281776, A281777, A281778, A281779, A281780.

Formula

a(n) = 1/2*4! Stirling2(n, 4) + 9/2*5! Stirling2(n, 5) + 16*6! Stirling2(n, 6) + 295/12*7! Stirling2(n, 7) + 85/4*8! Stirling2(n, 8) + 49/4*9! Stirling2(n, 9) + 9/2*10! Stirling2(n, 10) + 11!*Stirling2(n, 11).
Showing 1-7 of 7 results.

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