A016311 -id:A016311 - cp's OEIS frontend

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

0, 2, 15, 80, 375, 1652, 7035, 29360, 120975, 494252, 2007555, 8120840, 32753175, 131818052, 529680075, 2125927520, 8525298975, 34165897052, 136857560595, 548011897400, 2193792030375, 8780400395252, 35137296305115, 140596265198480

Offset: 1

N. J. A. Sloane

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 intersecting but for which x is not a subset of y and y is not a subset of x, or 1) 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 10 2008

Graph theory formulation. Let P(A) be the power set of an n-element set A. Then a(n) = the number of edges in the intersection graph G of P(A). The vertices of G are the elements of P(A) and the edges of G are the pairs of elements {x,y} of P(A) such that x and y are intersecting (and x <> y). - Ross La Haye, Dec 23 2017

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

T. D. Noe, Table of n, a(n) for n = 1..200
Taylor Brysiewicz, Holger Eble, and Lukas Kühne, Enumerating chambers of hyperplane arrangements with symmetry, arXiv:2105.14542 [math.CO], 2021.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, in Russian, Diskretnaya Matematika, 11 (1999), no. 4, 127-138.
V. Jovovic and G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, English translation, in Discrete Mathematics and Applications, 9, (1999), no. 6.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A036240, A051180-A051185, A028243, A032263.

Cf. A003462, A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{10,-35,50,-24},{0,2,15,80},40] (* or *) With[{c=1/2!}, Table[ c(4^n-3^n-2^n+1),{n,40}]] (* Harvey P. Dale, May 11 2011 *)

a(n)=(4^n-3^n-2^n+1)/2 \\ Charles R Greathouse IV, Jul 25 2011

[(4^n - 2^n)/2-(3^n - 1)/2 for n in range(1,24)] # Zerinvary Lajos, Jun 05 2009

a(n) = (1/2) * (4^n - 3^n - 2^n + 1).
a(n) = 3*Stirling2(n+1,4) + 2*Stirling2(n+1,3). - Ross La Haye, Jan 10 2008
a(n) = A006516(n) - A003462(n). - Zerinvary Lajos, Jun 05 2009
From Harvey P. Dale, May 11 2011: (Start)
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4); a(0)=0, a(1)=2, a(2)=15, a(3)=80.
G.f.: x^2*(2-5*x)/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4). (End)
E.g.f.: exp(x)*(exp(x) - 1)^2*(exp(x) + 1)/2. - Stefano Spezia, Jun 26 2022

A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

Original entry on oeis.org

1, 15, 157, 1419, 11869, 94731, 733069, 5551323, 41378557, 304766187, 2224062061, 16112628987, 116053574365, 831966057483, 5941308640333, 42294437942811, 300292730428093, 2127439102098219, 15044413649559085

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (15,-68,84).

Cf. A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

[ n eq 1 select 1 else n eq 2 select 15 else n eq 3 select 157 else 15*Self(n-1)-68*Self(n-2) +84*Self(n-3): n in [1..20] ]; // Vincenzo Librandi, Aug 25 2011

CoefficientList[Series[1/((1-2x)(1-6x)(1-7x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{15, -68, 84}, {1, 15, 157}, 30]

Vec(1/((1-2*x)*(1-6*x)*(1-7*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(7^n - 2^n)/5-(6^n - 2^n)/4 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (7^(n+2) - 2^(n+2))/5-(6^(n+2) - 2^(n+2))/4. - Zerinvary Lajos, Jun 05 2009 [corrected by Joerg Arndt, Aug 25 2011]
From Vincenzo Librandi, Aug 25 2011: (Start)
a(n) = 15*a(n-1) - 68*a(n-2) + 84*a(n-3) for n > 2;
a(n) = 13*a(n-1) - 42*a(n-2) + 2^n for n > 1. (End)
E.g.f.: exp(2*x)*(1 - 45*exp(4*x) + 49*exp(5*x))/5. - Stefano Spezia, Aug 25 2025

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 11, 83, 535, 3171, 17871, 97483, 520055, 2731091, 14179231, 72992283, 373347975, 1900290211, 9635660591, 48715157483, 245723238295, 1237206060531, 6220389909951, 31239388241083, 156746696495015, 785932504682051, 3938458614335311, 19727477439571083

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (11,-38,40).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

CoefficientList[ Series[ 1/((1 - 2x)(1 - 4x)(1 - 5x)), {x, 0, 20} ], x ]
LinearRecurrence[{11,-38,40},{1,11,83},30] (* Harvey P. Dale, Nov 29 2022 *)

Vec(1/((1-2*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(5^n - 2^n)/3-(4^n - 2^n)/2 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = (2/3)*2^n - 8*(4)^n + (25/3)*5^n. - Antonio Alberto Olivares, May 12 2012

A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

Original entry on oeis.org

1, 13, 117, 905, 6461, 43953, 289717, 1868425, 11861421, 74423393, 462815717, 2858273145, 17556537181, 107373722833, 654414852117, 3977351721065, 24118423433741, 145982106270273, 882250466222917

Offset: 0

N. J. A. Sloane

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-52,60).

Cf. A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

LinearRecurrence[{13,-52,60},{1,13,117},20] (* Harvey P. Dale, Mar 26 2016 *)

[(6^n - 2^n)/4-(5^n - 2^n)/3 for n in range(2,21)] # Zerinvary Lajos, Jun 05 2009

a(n) = A016129(n+1) - A016127(n+1). - Zerinvary Lajos, Jun 05 2009
a(n) = 13*a(n-1) - 52*a(n-2) + 60*a(n-3), n >= 3.
a(n) = 11*a(n-1) - 30*a(n-2) + 2^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 7*a(n-1) - 10*a(n-2) + 6^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = 8*a(n-1) - 12*a(n-2) + 5^n, n >= 2. - Vincenzo Librandi, Mar 16 2011
a(n) = -5^(n+2)/3 + 9*6^n + 2^n/3. - R. J. Mathar, Mar 18 2011

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

Original entry on oeis.org

1, 25, 447, 6989, 101759, 1417941, 19180519, 253983853, 3309800367, 42599540357, 542895780791, 6863463633117, 86197420501375, 1076563471968373, 13382900349107463, 165700329729679181, 2044564737700501583, 25152545442794015589, 308625999807796411735, 3778261997130507936445

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (25,-178,264).

Cf. A006516, A016127, A016129, A016130, A016311, A016316, A016321, A016325. - Zerinvary Lajos, Jun 05 2009

Cf. A016135, A016136.

[(648*12^n +2^(n+1)-5*11^(n+2))/45 : n in [0..20]]; // Vincenzo Librandi, Oct 09 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 11 x) (1 - 12 x)), {x, 0, 15}], x] (* Michael De Vlieger, Jan 31 2018 *)

Vec(1/((1-2*x)*(1-11*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[(12^n - 2^n)/10-(11^n - 2^n)/9 for n in range(2,18)] # Zerinvary Lajos, Jun 05 2009

From Vincenzo Librandi, Oct 09 2011: (Start)
a(n) = (648*12^n + 2^(n+1) - 5*11^(n+2))/45.
a(n) = 23*a(n-1) - 132*a(n-2) + 2^n.
a(n) = 25*a(n-1) - 178*a(n-2) + 264*a(n-3), n >= 3. (End)
From Elmo R. Oliveira, Mar 26 2025: (Start)
E.g.f.: exp(2*x)*(648*exp(10*x) - 605*exp(9*x) + 2)/45.
a(n) = A016136(n+1) - A016135(n+1). (End)

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).

Original entry on oeis.org

1, 22, 313, 3666, 38493, 377286, 3529681, 31947322, 282198565, 2447183310, 20920905369, 176852694018, 1481626607917, 12322682753494, 101879323774177, 838170485025354, 6867569457133749, 56077266261254238

Offset: 0

N. J. A. Sloane

From Bruno Berselli, May 09 2013: (Start)
a(n) - 2*a(n-1), for n>0, gives A019928 (after 1);
a(n) - 5*a(n-1), for n>0, gives A016311 (after 1);
a(n) - 7*a(n-1), for n>0, gives A016297 (after 1);
a(n) - 8*a(n-1), for n>0, gives A016296 (after 1);
a(n) - 7*a(n-1) + 10*a(n-2), for n>1, gives A016177 (after 15);
a(n) - 9*a(n-1) + 14*a(n-2), for n>1, gives A016162 (after 13);
a(n) - 10*a(n-1) + 16*a(n-2), for n>1, gives A016161 (after 12);
a(n) - 12*a(n-1) + 35*a(n-2), for n>1, gives A016131 (after 10);
a(n) - 13*a(n-1) + 40*a(n-2), for n>1, gives A016130 (after 9);
a(n) - 15*a(n-1) + 56*a(n-2), for n>1, gives A016127 (after 7);
a(n) - 20*a(n-1) +131*a(n-2) -280*a(n-3), for n>2, gives A000079 (after 4);
a(n) - 17*a(n-1) +86*a(n-2) -112*a(n-3), for n>2, gives A000351 (after 25);
a(n) - 15*a(n-1) +66*a(n-2) -80*a(n-3), for n>2, gives A000420 (after 49);
a(n) - 14*a(n-1) +59*a(n-2) -70*a(n-3), for n>2, gives A001018 (after 64),
and naturally: a(n) - 22*a(n-1) + 171*a(n-2) - 542*a(n-3) + 560*a(n-4), for n>3, gives 0 (see Harvey P. Dale in Formula lines). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (22,-171,542,-560).

Cf. A000079, A000351, A000420, A001018, A016127, A016130, A016131, A016161, A016162, A016177, A016296, A016297, A016311, A019928.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!(1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)))); // Bruno Berselli, May 09 2013

CoefficientList[Series[1/((1-2x)(1-5x)(1-7x)(1-8x)),{x,0,30}],x] (* or *) LinearRecurrence[ {22,-171,542,-560},{1,22,313,3666},30] (* Harvey P. Dale, Jan 29 2013 *)

a(n) = n+=3; (5*8^n-9*7^n+5*5^n-2^n)/90 \\ Charles R Greathouse IV, Oct 03 2016

def A025992(n): return (5*pow(8,n+3)-9*pow(7,n+3)+pow(5,n+4)-pow(2,n+3))//90
print([A025992(n) for n in range(41)]) # G. C. Greubel, Dec 27 2024

a(0)=1, a(1)=22, a(2)=313, a(3)=3666, a(n) = 22*a(n-1) - 171*a(n-2) + 542*a(n-3) - 560*a(n-4). - Harvey P. Dale, Jan 29 2013
a(n) = (5*8^(n+3) - 9*7^(n+3) + 5^(n+4) - 2^(n+3))/90. - Yahia Kahloune, May 07 2013
E.g.f.: (1/90)*(-8*exp(2*x) + 625*exp(5*x) - 3087*exp(7*x) + 2560*exp(8*x)). - G. C. Greubel, Dec 27 2024

cp's OEIS Frontend

A036239 Number of 2-element intersecting families of an n-element set; number of 2-way interactions when 2 subsets of power set on {1..n} are chosen at random.

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A016304 Expansion of 1/((1-2*x)*(1-6*x)*(1-7*x)).

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A016282 Expansion of 1/((1-2*x)*(1-4*x)*(1-5*x)).

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A016295 Expansion of 1/((1-2x)(1-5x)(1-6x)).

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A016633 Expansion of g.f. 1/((1-2*x)*(1-11*x)*(1-12*x)).

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A025992 Expansion of 1/((1-2*x)*(1-5*x)*(1-7*x)*(1-8*x)).

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A016304 Expansion of 1/((1-2x)(1-6x)(1-7*x)).

A016282 Expansion of 1/((1-2x)(1-4x)(1-5*x)).

A016633 Expansion of g.f. 1/((1-2x)(1-11x)(1-12*x)).

A025992 Expansion of 1/((1-2x)(1-5x)(1-7x)(1-8*x)).