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 1-10 of 30 results. Next

A051185 Number of intersecting families of an n-element set. Also number of n-variable clique Boolean functions.

Original entry on oeis.org

2, 6, 40, 1376, 1314816, 912818962432, 291201248266450683035648, 14704022144627161780744368338695925293142507520, 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

Comments

Also the number of n-ary Boolean polymorphisms of the binary Boolean relation OR, namely the Boolean functions f(x1,...,xn) with the property that (x1 or y1) and ... and (xn or yn) implies f(x1,...,xn) or f(y1,...,yn). - Don Knuth, Dec 04 2019
These values are necessarily divisible by powers of 2. The sequence of exponents begins 1, 1, 3, 5, 12, 22, 49, 93, ... , 2^(n-1)-C(n-1,floor(n/2)-1), ... (cf. A191391). - Andries E. Brouwer, Aug 07 2012
a(1) = 2^1.
a(2) = 6 = 2^1 * 3
a(3) = 2^3 * 5.
a(4) = 2^5 * 43.
a(5) = 2^12 * 3 * 107.
a(6) = 2^22 * 13 * 16741.
a(7) = 2^49 * 2111 * 245039,
a(8) = 2^93 * 3^2 * 5 * 7211 * 76697 * 59656829,
a(9) = 2^200 * 1823 * 2063 * 576967 * 3600144350906020591.
An intersecting family is a collection of subsets of {1,2,...,n} such that the intersection of every subset with itself or with any other subset in the family is nonempty. The maximum number of subsets in an intersecting family is 2^(n-1). - Geoffrey Critzer, Aug 16 2013

Examples

			a(2) = 6 because we have: {}, {{1}}, {{2}}, {{1, 2}}, {{1}, {1, 2}}, {{2}, {1, 2}}. - _Geoffrey Critzer_, Aug 16 2013

References

  • V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).
  • Pogosyan G., Miyakawa M., A. Nozaki, Rosenberg I., The Number of Clique Boolean Functions, IEICE Trans. Fundamentals, Vol. E80-A, No. 8, pp. 1502-1507, 1997/8.

Links

Crossrefs

Cf. A036239, A051180-A051184.

Programs

  • Mathematica
    Table[Length[
  Select[Subsets[Subsets[Range[1, n]]],
   Apply[And,
     Flatten[Table[
       Table[Intersection[#[[i]], #[[j]]] != {}, {i, 1,
Length[#]}], {j, 1, Length[#]}]]] &]], {n, 1, 4}] (* Geoffrey Critzer, Aug 16 2013 *)

Extensions

a(8)-a(9) by Andries E. Brouwer, Aug 07 2012, Dec 11 2012

A051180 Number of 3-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 13, 222, 2585, 25830, 238833, 2111382, 18142585, 152937510, 1271964353, 10476007542, 85662034185, 696700867590, 5643519669073, 45575393343702, 367206720319385, 2953481502692070, 23723872215168993, 190372457332919862
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

Links

Crossrefs

Cf. A036239, A051181-A051185.

Programs

  • Maple
    seq(1/3!*(8^n-3*6^n+3*5^n-4*4^n+3*3^n+2*2^n-2),n=0..40);
  • Mathematica
    Table[1/3!(8^n-3*6^n+3*5^n-4*4^n+3*3^n+2*2^n-2),{n,0,30}] (* or *) LinearRecurrence[{29,-343,2135,-7504,14756,-14832,5760},{0,0,0,13,222,2585,25830},30] (* Harvey P. Dale, Jul 07 2013 *)
  • PARI
    for(n=0,25, print1((1/3!)*(8^n-3*6^n+3*5^n-4*4^n+3*3^n+2*2^n-2), ", ")) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = (1/3!)*(8^n - 3*6^n + 3*5^n - 4*4^n + 3*3^n + 2*2^n - 2).
G.f. x^3*(744*x^3 - 606*x^2 + 155*x - 13)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Jul 29 2012
a(0)=0, a(1)=0, a(2)=0, a(3)=13, a(4)=222, a(5)=2585, a(6)=25830, a(n) = 29*a(n-1) - 343*a(n-2) + 2135*a(n-3) - 7504*a(n-4) + 14756*a(n-5) - 14832*a(n-6) + 5760*a(n-7). - Harvey P. Dale, Jul 07 2013

Extensions

More terms from Sascha Kurz, Mar 25 2002

A051181 Number of 4-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 4, 365, 11770, 278455, 5715094, 108498285, 1963243930, 34404675635, 589459538734, 9933916068505, 165358097339890, 2726894329246815, 44648990949187174, 727080119853611525, 11790570902483264650, 190587735542474633995, 3073193346666282232414
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

Links

Crossrefs

Cf. A036239, A051180-A051185.

Programs

  • Mathematica
    Table[1/4! (16^n - 6*12^n + 12*10^n - 9^n - 22*8^n + 15*7^n + 12*6^n - 17*5^n + 17*4^n - 11*3^n - 6*2^n + 6), {n, 0, 50}] (* G. C. Greubel, Oct 06 2017 *)
LinearRecurrence[{83,-3052,65670,-919413,8804499,-58966886,277278100,-904270136,1982352768,-2749917312,2142305280,-696729600},{0,0,0,4,365,11770,278455,5715094,108498285,1963243930,34404675635,589459538734},20] (* Harvey P. Dale, Jul 04 2019 *)
  • PARI
    for(n=0,25, print1((1/4!)*(16^n-6*12^n+12*10^n-9^n-22*8^n+15*7^n +12*6^n-17*5^n+17*4^n-11*3^n-6*2^n+6), ", ")) \\ G. C. Greubel, Oct 06 2017

Formula

a(n) = (1/4!)*(16^n - 6*12^n + 12*10^n - 9^n - 22*8^n + 15*7^n + 12*6^n - 17*5^n + 17*4^n - 11*3^n - 6*2^n + 6).
G.f.: -x^3*(64667520*x^8 - 81966960*x^7 + 42070268*x^6 - 11421992*x^5 + 1766529*x^4 - 152845*x^3 + 6317*x^2 - 33*x - 4)/((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-1)*(10*x-1)*(12*x-1)*(16*x-1)). - Colin Barker, Jul 30 2012

Extensions

More terms from Harvey P. Dale, Jul 04 2019

A051184 Number of 7-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 80, 169125, 71102400, 18047221707, 3623784887164, 638772147728325, 103751227132038920, 15931275037246743999, 2348130220089143792148, 335520750110815538499945, 46803828588394634589433120
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Crossrefs

Cf. A036239, A051180-A051185.

Formula

1/7! (128^n - 21*96^n + 105*80^n - 35*72^n + 105*68^n - 42*66^n + 7*65^n - 476*64^n - 630*60^n + 1785*56^n + 315*54^n - 210*52^n - 105*51^n + 1260*50^n - 105*49^n - 1575*48^n - 2520*46^n - 105*45^n + 1638*44^n + 840*43^n - 6615*42^n + 1050*41^n + 4130*40^n - 1890*39^n + 14595*38^n + 2835*37^n - 7945*36^n - 1554*35^n - 18711*34^n - 12572*33^n + 24710*32^n + 4620*31^n + 560*30^n + 25995*29^n - 16905*28^n - 13545*27^n - 6510*26^n - 42945*25^n + 12005*24^n + 102011*23^n - 4648*22^n - 87780*21^n - 15785*20^n + 43120*19^n + 21364*18^n + 4200*17^n - 37205*16^n - 17105*15^n + 36386*14^n + 28644*13^n - 57603*12^n + 24150*11^n + 4585*10^n - 16289*9^n + 20943*8^n - 12754*7^n - 287*6^n + 4137*5^n - 3388*4^n + 1764*3^n + 720*2^n - 720)

A053154 Number of 2-element intersecting families (with not necessarily distinct sets) of an n-element set.

Original entry on oeis.org

0, 1, 5, 22, 95, 406, 1715, 7162, 29615, 121486, 495275, 2009602, 8124935, 32761366, 131834435, 529712842, 2125993055, 8525430046, 34166159195, 136858084882, 548012945975, 2193794127526, 8780404589555, 35137304693722
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

Keywords

Comments

Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are intersecting but for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x. - Ross La Haye, Jan 11 2008

Links

Crossrefs

Cf. A036239, A083324.
Cf. A000225, A032263, A028243.

Programs

  • Magma
    [(4^n-3^n+2^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Oct 06 2017
  • Mathematica
    Table[(4^n-3^n+2^n-1)/2, {n,1,30}] (* Clark Kimberling, Mar 12 2012 *)
CoefficientList[Series[x (1 - 5 x + 7 x^2) / ((1 - x) (1 - 4 x) (1 - 3 x) (1 - 2 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Oct 06 2017 *)
  • PARI
    a(n) = (4^n-3^n+2^n-1)/2; \\ Michel Marcus, Nov 30 2015

Formula

a(n) = (A083324(n) - 1)/2.
a(n) = (4^n - 3^n + 2^n - 1)/2.
a(n) = 3*StirlingS2(n+1,4) + 2*StirlingS2(n+1,3) + StirlingS2(n+1,2). - Ross La Haye, Jan 11 2008
From Wolfdieter Lang, Oct 28 2011 (Start)
E.g.f.: Sum_{j=1..4} ((-1)^j*exp(j*x))/2 = exp(x)*(exp(4*x)-1)/(exp(x)+1)/2.
O.g.f.: Sum_{j=1..4} (((-1)^j)/(1-j*x))/2 = x*(1-5*x+7*x^2)/product(1-j*x,j=1..4). See A196847.
(End)
G.f.: x*(1-5*x+7*x^2)/((1-x)*(1-4*x)*(1-3*x)*(1-2*x)). - Vincenzo Librandi, Oct 06 2017

A036240 Number of 3-way interactions when 3 subsets of power set on {1..n} are chosen at random; number of Boolean functions of n variables and rank 3 from Post class F(8,inf).

Original entry on oeis.org

0, 0, 12, 200, 2280, 22420, 205212, 1806000, 15522960, 131383340, 1100093412, 9138243400, 75445046040, 619838752260, 5072272077612, 41371548418400, 336519691295520, 2730963319321180, 22119245290765812, 178854325039467000, 1444135501669535400
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • W. W. Kokko, "Interactions", manuscript, 1983.

Links

Crossrefs

Cf. A036239.

Programs

  • Magma
    [(8^n-7^n-3*4^n+3*3^n+2*2^n-2)/6 : n in [1..30]]; // Wesley Ivan Hurt, Oct 23 2014
  • Maple
    A036240:=n->(8^n-7^n-3*4^n+3*3^n+2*2^n-2)/6: seq(A036240(n), n=1..30); # Wesley Ivan Hurt, Oct 23 2014
  • Mathematica
    CoefficientList[Series[4 x^2 (43 x^2 - 25 x + 3)/((x - 1) (2 x - 1) (3 x - 1) (4 x - 1) (7 x - 1) (8 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 21 2013 *)
LinearRecurrence[{25,-241,1135,-2734,3160,-1344},{0,0,12,200,2280,22420},30] (* Harvey P. Dale, Dec 29 2013 *)
  • PARI
    a(n) = (1/3!)*(8^n-7^n-3*4^n+3*3^n+2*2^n-2); \\ Joerg Arndt, Oct 21 2013

Formula

a(n) = (8^n-7^n-3*4^n+3*3^n+2*2^n-2)/6.
G.f.: 4*x^3*(43*x^2-25*x+3) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(7*x-1)*(8*x-1)). - Colin Barker, Dec 10 2012
a(n) = 25*a(n-1)-241*a(n-2)+1135*a(n-3)-2734*a(n-4)+3160*a(n-5)-1344*a(n-6). - Wesley Ivan Hurt, Oct 23 2014
E.g.f.: exp(x)*(exp(x) - 1)^3*(exp(x) + 1)^2*(exp(2*x) + 2)/6. - Stefano Spezia, Jul 29 2022

A051182 Number of 5-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 371, 38163, 2236504, 103998636, 4289058501, 164693276181, 6034793020298, 213993130915542, 7407880110115111, 251837583669470799, 8443568934653875932, 280082506996725346368
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Crossrefs

Cf. A036239, A051180-A051185.

Formula

1/5!(32^n-10*24^n+30*20^n-5*18^n+5*17^n-80*16^n-30*15^n+135*14^n+30*13^n-80*12^n-2*11^n+10*10^n-100*9^n+240*8^n-160*7^n-44*6^n+95*5^n-85*4^n+50*3^n+24*2^n-24).

A051183 Number of 6-element intersecting families of an n-element set.

Original entry on oeis.org

0, 0, 0, 0, 230, 91993, 14037879, 1509286261, 136653987232, 11209147489701, 862949794999193, 63573922606869037, 4535012297248660194, 315713834759742768349, 21570075957885603579067
Offset: 0

Views

Author

Vladeta Jovovic, Goran Kilibarda

Keywords

References

  • V. Jovovic, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, Diskretnaya Matematika, 11 (1999), no. 4, 127-138 (translated in Discrete Mathematics and Applications, 9, (1999), no. 6).

Crossrefs

Cf. A036239, A051180-A051185.

Formula

1/6! (64^n - 15*48^n + 60*40^n - 15*36^n + 30*34^n - 6*33^n - 215*32^n - 180*30^n + 585*28^n + 45*27^n + 60*26^n + 150*25^n - 510*24^n - 360*23^n + 168*22^n - 585*21^n + 795*20^n + 1665*19^n - 1890*18^n - 2175*17^n + 3305*16^n + 1775*15^n - 3795*14^n - 870*13^n + 3123*12^n - 1075*11^n - 495*10^n + 1460*9^n - 2245*8^n + 1424*7^n + 150*6^n - 590*5^n + 499*4^n - 274*3^n - 120*2^n + 120)

A053152 Number of 2-element intersecting families whose union is an n-element set.

Original entry on oeis.org

0, 2, 9, 32, 105, 332, 1029, 3152, 9585, 29012, 87549, 263672, 793065, 2383292, 7158069, 21490592, 64504545, 193579172, 580868589, 1742867912, 5229128025, 15688432652, 47067395109, 141206379632, 423627527505, 1270899359732, 3812731633629, 11438262009752
Offset: 1

Views

Author

Vladeta Jovovic, Goran Kilibarda, Feb 28 2000

Keywords

Links

Crossrefs

Cf. A036239, A064686 (first differences).

Programs

  • Magma
    [Floor((3^n-2^n)/2): n in [1..30]]; // Vincenzo Librandi, Mar 17 2015
  • Maple
    A053152:=n->floor((3^n-2^n)/2): seq(A053152(n), n=1..30); # Wesley Ivan Hurt, Mar 19 2015
  • Mathematica
    CoefficientList[Series[x (2 - 3 x) / ((1 - x) (1 - 2 x) (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 17 2015 *)
LinearRecurrence[{6,-11,6}, {0,2,9}, 50] (* G. C. Greubel, Oct 06 2017 *)
  • PARI
    for(n=1,50, print1((1/2)*(3^n -2^n -1), ", ")) \\ G. C. Greubel, Oct 06 2017
  • Sage
    [(3^n - 1)/2-2^(n-1) for n in range(1,27)] # Zerinvary Lajos, Jun 05 2009

Formula

a(n) = (1/2!)*(3^n-2^n-1).
From Colin Barker, Jun 26 2012: (Start)
a(n) = 6*a(n-1)-11*a(n-2)+6*a(n-3).
G.f.: x^2*(2-3*x)/((1-x)*(1-2*x)*(1-3*x)). (End)
a(n) = floor((3^n-2^n)/2). - Wesley Ivan Hurt, Mar 16 2015

Extensions

More terms from James Sellers, Mar 01 2000
a(27)-a(28) from Vincenzo Librandi, Mar 17 2015

A053156 Number of 2-element intersecting families (with not necessarily distinct sets) whose union is an n-element set.

Original entry on oeis.org

1, 3, 10, 33, 106, 333, 1030, 3153, 9586, 29013, 87550, 263673, 793066, 2383293, 7158070, 21490593, 64504546, 193579173, 580868590, 1742867913, 5229128026, 15688432653, 47067395110, 141206379633, 423627527506, 1270899359733
Offset: 1

Views

Author

Vladeta Jovovic and Goran Kilibarda, Feb 28 2000

Keywords

Comments

Let P(A) be the power set of an n-element set A. Then a(n+1) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are disjoint and for which either x is a subset of y or y is a subset of x, or 1) x and y are disjoint and for which x is not a subset of y and y is not a subset of x, or 2) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 3) x = y. - Ross La Haye, Jan 12 2008
From Paul Barry, Apr 27 2003: (Start)
With offset 0, this is a(n) = (3*3^n - 2*2^n + 1)/2.
G.f. (1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f. (3*exp(3*x) - 2*exp(2*x) + exp(x))/2.
Binomial transform of A083329.
Second binomial transform of A040001. (End)

Links

Crossrefs

Cf. A000225, A000392, A028243, A000079.
Cf. A036239.
Column k=2 of A288638.
Third column of A294201.

Programs

  • Magma
    [(3^n-2^n+1)/2: n in [1..30]]; // G. C. Greubel, Oct 06 2017
  • Maple
    A053156:=n->(3^n - 2^n + 1)/2: seq(A053156(n), n=1..40); # Wesley Ivan Hurt, Oct 06 2017
  • Mathematica
    LinearRecurrence[{6,-11,6}, {1, 3, 10}, 50] (* or *) Table[(3^n - 2^n + 1)/2, {n,1,50}] (* G. C. Greubel, Oct 06 2017 *)
  • PARI
    a(n) = (3^n-2^n+1)/2; \\ Michel Marcus, Nov 30 2015

Formula

a(n) = (3^n - 2^n + 1)/2.
a(n) = StirlingS2(n+2,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 12 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
G.f.: x*(1-3*x+3*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)
Showing 1-10 of 30 results. Next

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

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