A051588 -id:A051588 - cp's OEIS frontend

A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

1, 15, 151, 1275, 9751, 70035, 481951, 3216795, 20991751, 134667555, 852639151, 5343198315, 33212784151, 205111785075, 1260114546751, 7708980203835, 46999640806951, 285743822630595, 1733261544204751

Offset: 0

Robert G. Wilson v

2*a(n-2) = 6^n - 2*5^n + 4^n is the number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) first and third row have a common 1, (c) second and third row have no common 1. - Andi Fugard and Vladeta Jovovic, Jul 26 2008

This is the third column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See A193685 for general comments. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
Index entries for linear recurrences with constant coefficients, signature (15,-74,120).

Cf. A051588, A000302, A005060, A003468.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-5*x)*(1-6*x)))); // Vincenzo Librandi, Jun 24 2013

I:=[1, 15, 151]; [n le 3 select I[n] else 15*Self(n-1)-74*Self(n-2)+120*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 5 x) (1 - 6 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

Vec(1/((1-4*x)*(1-5*x)*(1-6*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2^(3 + 2*n) + 2^(1 + n) * 3^(2 + n) - 5^(2 + n). - Andi Fugard, Jul 22 2008
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,4), n >= 2. - Milan Janjic, Apr 26 2009
O.g.f.: 1/((1-4*x)*(1-5*x)*(1-6*x)).
E.g.f.: (d^2/dx^2)(exp(4*x)*((exp(x)-1)^2)/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 08 2011
a(n) = 15*a(n-1) - 74*a(n-2) + 120*a(n-3). - Vincenzo Librandi, Jun 24 2013

A051589 Number of 5xn binary matrices such that any 2 rows have a common 1.

Original entry on oeis.org

0, 1, 63, 3367, 167835, 7803391, 339133803, 13887495007, 541044196875, 20237096702431, 732455240043243, 25820836854042847, 891331324715015115, 30260208833985800671, 1013882831306569043883, 33620617443978687281887, 1105857774681062127612555

Offset: 0

Vladeta Jovovic, Goran Kilibarda

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..670
V. Jovovic, 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, G. Kilibarda, On the number of Boolean functions in the Post classes F^{mu}_8, (English translation), Discrete Mathematics and Applications, 9, (1999), no. 6.

Cf. A005061, A051587, A051588.

List([0..20], n-> 32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n); # G. C. Greubel, Nov 12 2019

[32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n: n in [0..20]]; // Vincenzo Librandi, Sep 18 2018

A051589(n):=32^n -10*24^n +30*20^n -5*18^n +5*17^n -70*16^n -30*15^n + 135*14^n +30*13^n -140*12^n -2*11^n +130*10^n -110*9^n +45*8^n -10*7^n +6^n; seq(A051589(n), n=0..20); # G. C. Greubel, Nov 12 2019

Table[32^n -10*24^n +30*20^n -5*18^n +5*17^n -70*16^n -30*15^n +135*14^n +30*13^n -140*12^n -2*11^n +130*10^n -110*9^n +45*8^n -10*7^n +6^n, {n, 0, 30}] (* Vincenzo Librandi, Sep 18 2018 *)

vector(21, n, m=n-1; 32^m -10*24^m +30*20^m -5*18^m +5*17^m -70*16^m -30*15^m +135*14^m +30*13^m -140*12^m -2*11^m +130*10^m -110*9^m +45*8^m -10*7^m +6^m) \\ G. C. Greubel, Nov 12 2019

[32^n-10*24^n+30*20^n-5*18^n+5*17^n-70*16^n-30*15^n +135*14^n +30*13^n-140*12^n-2*11^n+130*10^n-110*9^n+45*8^n-10*7^n +6^n for n in (0..20)] # G. C. Greubel, Nov 12 2019

a(n) = 32^n - 10*24^n + 30*20^n - 5*18^n + 5*17^n - 70*16^n - 30*15^n + 135*14^n + 30*13^n - 140*12^n - 2*11^n + 130*10^n - 110*9^n + 45*8^n - 10*7^n + 6^n.

G.f.: x*(933561925632000*x^14 -1286309121638400*x^13 +786606914672640*x^12 -287219252934144*x^11 +70324589076096*x^10 -12248067009984*x^9 +1568017231256*x^8 -150181430252*x^7 +10834851518*x^6 -587198697*x^5 +23594853*x^4 -684354*x^3 +13636*x^2 -169*x +1) / ((6*x -1)*(7*x -1)*(8*x -1)*(9*x -1)*(10*x -1)*(11*x -1)*(12*x -1)*(13*x -1)*(14*x -1)*(15*x -1)*(16*x -1)*(17*x -1)*(18*x -1)*(20*x -1)*(24*x -1)*(32*x -1)). - Colin Barker, Feb 22 2013

Revised Aug 03 2000

A052387 Number of 3 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

Original entry on oeis.org

0, 1, 8, 37, 127, 358, 876, 1926, 3894, 7359, 13156, 22451, 36829, 58396, 89896, 134844, 197676, 283917, 400368, 555313, 758747, 1022626, 1361140, 1791010, 2331810, 3006315, 3840876, 4865823, 6115897, 7630712, 9455248

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Mar 11 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000
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).
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A051588, A051587, A051589.

[n*(n+1)*(n+2)*(n+3)*(n^3+22*n^2+53*n+134)/5040: n in [0..30]]; // Wesley Ivan Hurt, May 15 2014

A052387:=n->n*(n+1)*(n+2)*(n+3)*(n^3+22*n^2+53*n+134)/5040; seq(A052387(n), n=0..30); # Wesley Ivan Hurt, May 15 2014

Table[n*(n + 1)*(n + 2)*(n + 3)*(n^3 + 22*n^2 + 53*n + 134)/5040, {n,
0, 30}] (* Wesley Ivan Hurt, May 15 2014 *)

x='x+O('x^50); concat([0], Vec(-x*(x^3-x^2-1)/(x-1)^8)) \\ G. C. Greubel, Oct 07 2017

a(n) = n*(n+1)*(n+2)*(n+3)*(n^3 +22*n^2 +53*n +134)/5040.

G.f.: -x*(x^3-x^2-1)/(x-1)^8. - Colin Barker, Nov 05 2012

A052388 Number of 4 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

Original entry on oeis.org

0, 1, 16, 146, 955, 4905, 20907, 76851, 250530, 739612, 2009177, 5085119, 12109526, 27348478, 58955082, 121956402, 243172488, 469115187, 878387366, 1600751976, 2845918041, 4946262815, 8419256605, 14057377245, 23055913530, 37192403430, 59075703351, 92488040301

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Mar 11 2000

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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1).

Cf. A051588, A051587, A051589.

[n*(n+1)*(n+2)*(n+3)*(n+4)*(n^10 +110*n^9 +5445*n^8 +160050*n^7 +2906463*n^6 +30644250*n^5 +176659055*n^4 +711220750*n^3 +1781493036*n^2 +4034382840*n +4159814400)/1307674368000: n in [0..25]]; // G. C. Greubel, Oct 07 2017

CoefficientList[Series[-x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16, {x, 0, 50}], x] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0], Vec(-x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16)) \\ G. C. Greubel, Oct 07 2017

a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(n^10 +110*n^9 +5445*n^8 +160050*n^7 +2906463*n^6 +30644250*n^5 +176659055*n^4 +711220750*n^3 +1781493036*n^2 +4034382840*n +4159814400)/1307674368000.

G.f.: -x*(x^10 -5*x^9 +10*x^8 -14*x^7 +21*x^6 -19*x^5 -5*x^4 +21*x^3 -10*x^2 -1)/(x-1)^16. - Colin Barker, Nov 05 2012

A140961 Number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) second and third row have a common 1.

Original entry on oeis.org

1, 17, 205, 2129, 20341, 184457, 1615405, 13808609, 116015461, 962575097, 7913168605, 64610052689, 524855128981, 4247421698537, 34274519697805, 275985344786369, 2218709434248901, 17815093293410777, 142915542082163005, 1145704555158361649, 9179974972732223221

Offset: 1

Andi Fugard, Vladeta Jovovic, Jul 26 2008

Andi Fugard, Counting first-order models (with n individuals) of syllogisms.
Index entries for linear recurrences with constant coefficients, signature (19,-118,240).

Cf. A051588.

a(n) = 8^n-2*6^n+5^n.

G.f.: x*(2*x-1)/((5*x-1)*(6*x-1)*(8*x-1)). [Colin Barker, Nov 05 2012]

A319366 Number of 6 X n binary matrices such that any 2 rows have a common 1.

Original entry on oeis.org

1, 127, 14197, 1527655, 154708741, 14581420567, 1282928605477, 106281575400295, 8370106554738181, 632240233746846007, 46159332156459328357, 3278558540783856976135, 227767526682511220042821, 15545657368091391819871447, 1046175606578621216182684837

Offset: 1

T. V. Raziman, Sep 17 2018

nonn

Mathematics Stack Exchange, An interesting combinatorics problem
Quora, How many ways are there to set N people into M groups so that every two groups have at least one common member?

Cf. A051587, A051588, A051589.

a(n) = 64^n - 15*48^n + 60*40^n - 15*36^n + 30*34^n - 6*33^n - 200*32^n - 180*30^n + 585*28^n + 45*27^n + 60*26^n + 150*25^n - 660*24^n - 360*23^n + 168*22^n - 585*21^n + 1245*20^n + 1665*19^n - 1965*18^n - 2100*17^n + 2170*16^n + 1325*15^n - 1770*14^n - 420*13^n + 1533*12^n - 1105*11^n + 435*10^n - 105*9^n + 15*8^n - 7^n (proved in the Quora answer).

cp's OEIS Frontend

A016103 Expansion of 1/((1-4x)(1-5x)(1-6x)).

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A051589 Number of 5xn binary matrices such that any 2 rows have a common 1.

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A052387 Number of 3 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

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A052388 Number of 4 X n binary matrices such that any 2 rows have a common 1, up to column permutations.

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A140961 Number of 3 X n {0,1}-matrices such that: (a) first and second row have a common 1, (b) second and third row have a common 1.

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A319366 Number of 6 X n binary matrices such that any 2 rows have a common 1.

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