A016269 -id:A016269 - cp's OEIS frontend

A094037 Number of connected 6-element antichains on a labeled n-set.

0, 0, 0, 0, 1, 1345, 738741, 185165477, 29458046177, 3541242666045, 354515664467077, 31326419674855789, 2535191648955942273, 192567615994193565125, 13962461827318220986133, 978010022290154153870661

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

Cf. A016269, A047707, A051112-A051118, A094033-A094037.

E.g.f.: (exp(63*x) - 30*exp(47*x) + 120*exp(39*x) + 60*exp(35*x) + 60*exp(33*x) - 18*exp(32*x) - 339*exp(31*x) - 720*exp(29*x) + 810*exp(27*x) + 120*exp(26*x) + 480*exp(25*x) + 480*exp(24*x) - 600*exp(23*x) - 720*exp(22*x) - 240*exp(21*x) - 900*exp(20*x) + 1740*exp(19*x) + 615*exp(18*x) + 180*exp(17*x) + 435*exp(16*x) - 1445*exp(15*x) - 3270*exp(14*x) + 1710*exp(13*x) + 4620*exp(12*x) - 3360*exp(11*x) - 3210*exp(10*x) + 3360*exp(9*x) + 6810*exp(8*x) - 12465*exp(7*x) + 5985*exp(6*x) + 7110*exp(5*x) - 18555*exp(4*x) + 17884*exp(3*x) - 8352*exp(2*x) + 1764*exp(x) - 120)/6!.

A084883 Number of (k,m,n)-multiantichains of multisets with k=3 and m=6.

Original entry on oeis.org

1, 3, 64, 8022, 6822072, 14068794534, 26314469636622, 37310026340520678, 42667193588371160460, 42169580808988409450310, 37803058273249518925923210, 31733179110752959606870643334

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

nonn

By a (k,m,n)-multiantichain of multisets we mean an m-multiantichain of k-bounded multisets on an n-set. The elements of a multiantichain could have the multiplicities greater than 1. A multiset is called k-bounded if every its element has the multiplicity not greater than k-1.

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.

Cf. A016269, A047707, A051112-A051118, A084869-A084882.

a(n) = (1/6!)*(729^n - 30*486^n + 120*378^n + 60*324^n + 60*294^n - 360*279^n - 12*276^n - 720*252^n + 45*243^n + 90*234^n + 720*231^n + 120*216^n + 720*210^n - 240*205^n + 360*196^n - 720*189^n - 180*187^n + 720*186^n - 720*176^n + 120*168^n - 720*167^n + 360*165^n - 900*162^n - 720*157^n + 180*156^n + 720*148^n - 240*145^n + 720*138^n + 30*134^n - 240*129^n + 2700*126^n - 360*120^n + 180*111^n + 900*108^n - 20*102^n + 450*98^n - 5400*93^n - 5400*84^n + 685*81^n + 1350*78^n + 5400*77^n + 5400*70^n - 5400*63^n + 900*56^n - 8220*54^n + 16440*42^n + 2740*36^n - 16440*31^n + 4275*27^n + 4110*26^n - 25650*18^n + 25650*14^n + 10474*9^n - 20948*6^n + 7560*3^n).

A069403 a(n) = 2Fibonacci(2n+1) - 1.

Original entry on oeis.org

1, 3, 9, 25, 67, 177, 465, 1219, 3193, 8361, 21891, 57313, 150049, 392835, 1028457, 2692537, 7049155, 18454929, 48315633, 126491971, 331160281, 866988873, 2269806339, 5942430145, 15557484097, 40730022147, 106632582345, 279167724889, 730870592323, 1913444052081

Offset: 0

R. H. Hardin, Mar 22 2002

Half the number of n X 3 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Indices of A017245 = 9*n + 7 = 7, 16, 25, 34, for submitted A153819 = 16, 34, 88,. A153819(n) = 9*a(n) + 7 = 18*F(2*n+1) -2; F(n) = Fibonacci = A000045, 2's = A007395. Other recurrence: a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3). - Paul Curtz, Jan 02 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Yurii S. Bystryk, Vitalii L. Denysenko, and Volodymyr I. Ostryk, Lune and Lens Sequences, ResearchGate preprint, 2024. See pp. 41, 56.
J. Hietarinta and C.-M. Viallet, Singularity confinement and chaos in discrete systems, Physical Review Letters 81 (1998), pp. 326-328.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Cf. A000045, A084707.
Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
Equals A052995 - 1.
Bisection of A001595, A062114, A066983.

List([0..30], n-> 2*Fibonacci(2*n+1)-1); # G. C. Greubel, Jul 11 2019

[2*Fibonacci(2*n+1)-1: n in [0..30]]; // Vincenzo Librandi, Apr 18 2011

a[n_]:= a[n] = 3a[n-1] - 3a[n-3] + a[n-4]; a[0] = 1; a[1] = 3; a[2] = 9; a[3] = 25; Table[ a[n], {n, 0, 30}]
Table[2*Fibonacci[2*n+1]-1, {n,0,30}] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{4,-4,1},{1,3,9},30] (* Harvey P. Dale, Sep 22 2020 *)

a(n) = 2*fibonacci(2*n+1)-1 \\ Charles R Greathouse IV, Jun 11 2015

Vec((1-x+x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016

[2*fibonacci(2*n+1)-1 for n in (0..30)] # G. C. Greubel, Jul 11 2019

a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 25; a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4).
a(n) = 3*a(n-1) - a(n-2) + 1 for n>1, a(1) = 3, a(0) = 0. - Reinhard Zumkeller, May 02 2006
From R. J. Mathar, Feb 23 2009: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
G.f.: (1-x+x^2)/((1-x)*(1-3*x+x^2)). (End)
a(n) = 1 + 2*Sum_{k=0..n} Fibonacci(2*k) = 1+2*A027941(n). - Gary Detlefs, Dec 07 2010
a(n) = (2^(-n)*(-5*2^n -(3-sqrt(5))^n*(-5+sqrt(5)) +(3+sqrt(5))^n*(5+sqrt(5))))/5. - Colin Barker, Nov 02 2016

Simpler definition from Vladeta Jovovic, Mar 19 2003

A025211 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).

Original entry on oeis.org

1, 14, 125, 910, 5901, 35574, 204205, 1132670, 6129101, 32566534, 170691885, 885423630, 4556561101, 23305343894, 118631189165, 601616805790, 3042056477901, 15346559343654, 77279066272045, 388583895311150, 1951684190615501, 9793511186181814, 49108010998116525

Offset: 0

N. J. A. Sloane

This gives the fourth column of the Sheffer triangle A143494 (2-restricted Stirling2 numbers). See the e.g.f. given below, and comments on the general case under A193685. - Wolfdieter Lang, Oct 08 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (14,-71,154,-120)

Cf. A000079, A001047, A016269.

[3^(n+3)/2 -2*4^(n+2)-2^(n+2)/3+5^(n+3)/6: n in [0..30]]; // Vincenzo Librandi, Jun 21 2011

CoefficientList[Series[1/((1 - 2 x) (1 - 3 x) (1 - 4 x) (1 - 5 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{14,-71,154,-120},{1,14,125,910},30] (* Harvey P. Dale, Feb 05 2020 *)

a(n)=n-=2;3^n*3/2-2*4^n-2^n/3+5^n*5/6 \\ Charles R Greathouse IV, Jun 21 2011

If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = (-1)^(n-1)*f(n,3,-5), (n >= 3). - Milan Janjic, Apr 26 2009
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*stirling2(k,j)*x^(m-k) then a(n-3) = f(n,3,2), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 3^(n+3)/2 - 2*4^(n+2) - 2^(n+2)/3 + 5^(n+3)/6. - R. J. Mathar, Mar 22 2011
O.g.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: (d^3/dx^3) (exp(2*x)*((exp(x)-1)^3)/3!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011

A051113 Number of monotone Boolean functions of n variables with 5 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 6, 2146, 304752, 25400564, 1557306954, 78817977462, 3513106214484, 143429796694888, 5501383287745422, 201652447559180618, 7148287976359243896, 247151326758617289372, 8386495692534098616210, 280574309728711561269214, 9286566498536162168164188

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112-A051118.

a(n) = 1/5! * (32^n-20 * 24^n+ 60 * 20^n+ 20 * 18^n+ 10 * 17^n-110 * 16^n-120 * 15^n+ 150 * 14^n+ 120 * 13^n-240 * 12^n+ 20 * 11^n+ 240 * 10^n+ 40 * 9^n-205 * 8^n+ 60 * 7^n-210 * 6^n+ 210 * 5^n+ 50 * 4^n-100 * 3^n+ 24 * 2^n).

G.f.: -2*x^4*(140561100029952000*x^15 -73258140662784000*x^14 -8396658614522880*x^13 +15284070825850368*x^12 -4918391338514880*x^11 +748203166795520*x^10 -45197506544400*x^9 -3280961201664*x^8 +887950976060*x^7 -80597007540*x^6 +3942400065*x^5 -98697251*x^4 +532770*x^3 +26970*x^2 -335*x -3) / ((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)*(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, Jul 14 2013

More terms from Colin Barker, Jul 14 2013

A051114 Number of monotone Boolean functions of n variables with 6 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1380, 759457, 192504214, 31169837405, 3827970163920, 392135190780649, 35468973527445018, 2937270598777421269, 228156280366446932500, 16904255174464832812001, 1208995011493806361868862, 84197134590686932418878093, 5746616155270206518199693720

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
V. Jovovic and G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

Colin Barker, Table of n, a(n) for n = 0..500
K. S. Brown, Dedekind's Problem
V. Jovovic, Illustration for A016269, A047707, A051112-A051118
V. Jovovic and 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).
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051115, A051116, A051117, A051118.

a(n) = (1/6!)*(64^n-30 * 48^n+ 120 * 40^n+ 60 * 36^n+ 60 * 34^n-12 * 33^n-345 * 32^n-720 * 30^n+ 810 * 28^n+ 120 * 27^n+ 480 * 26^n+ 360 * 25^n-480 * 24^n-720 * 23^n-240 * 22^n-540 * 21^n+ 1380 * 20^n+ 750 * 19^n+ 60 * 18^n-210 * 17^n-1535 * 16^n-1820 * 15^n+ 2250 * 14^n+ 1800 * 13^n-2820 * 12^n+ 300 * 11^n+ 2040 * 10^n+ 340 * 9^n-1815 * 8^n+ 510 * 7^n-1350 * 6^n+ 1350 * 5^n+ 274 * 4^n-548 * 3^n+ 120 * 2^n).

More terms from Colin Barker, Nov 26 2014

A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

Original entry on oeis.org

1, 8, 46, 230, 1066, 4718, 20266, 85310, 354106, 1455278, 5938186, 24104990, 97478746, 393095438, 1581931306, 6356390270, 25511588986, 102304505198, 409992599626, 1642294397150, 6576150108826, 26325519044558, 105364834103146, 421647614381630, 1687155299822266

Offset: 0

N. J. A. Sloane

a(n) is also the number of acyclic orientations of the complete bipartite graph K_{3,n}. - Vincent Pilaud, Sep 15 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..500
K. Imatomi, M. Kaneko, and E. Takeda, Multi-Poly-Bernoulli Numbers and Finite Multiple Zeta Values, J. Int. Seq. 17 (2014) # 14.4.5.
Ken Kamano, Sums of Products of Bernoulli Numbers, Including Poly-Bernoulli Numbers, J. Int. Seq. 13 (2010), 10.5.2.
Ken Kamano, Sums of Products of Poly-Bernoulli Numbers of Negative Index, Journal of Integer Sequences, Vol. 15 (2012), #12.1.3.
Masanobu Kaneko, Poly-Bernoulli numbers, Journal de théorie des nombres de Bordeaux, 9 no. 1 (1997), Pages 221-228.
Takao Komatsu, Some recurrence relations of poly-Cauchy numbers, J. Nonlinear Sci. Appl., (2019) Vol. 12, Issue 12, 829-845.
Index entries for sequences related to Bernoulli numbers.
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A027641, A027642, A027643, A027644, A027645, A027646, A027647, A027648, A027649, A027651.
First differences of A016269.
Row 3 of array A099594.

[6*4^n-6*3^n+2^n: n in [0..30]]; // Vincenzo Librandi, Jul 17 2011

a:= (n, k) -> (-1)^n*sum((-1)^j*j!*Stirling2(n, j)/(j+1)^k, j=0..n);
seq(a(n, -3), n = 0..30);

Table[6*4^n-6*3^n+2^n, {n,0,30}] (* G. C. Greubel, Feb 07 2018 *)

Vec((1-x)/((1-2*x)*(1-3*x)*(1-4*x)) + O(x^30)) \\ Michel Marcus, Feb 13 2015

[2^n -6*3^n +6*4^n for n in (0..30)] # G. C. Greubel, Aug 02 2022

a(n) = 6*4^n - 6*3^n + 2^n. - Vladeta Jovovic, Nov 14 2003
G.f.: (1-x)/((1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: exp(2*x) - 6*exp(3*x) + 6*exp(4*x). - G. C. Greubel, Aug 02 2022

A051115 Number of monotone Boolean functions of n variables with 7 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 490, 1308270, 1085660748, 483349680164, 147791677696350, 35419166732721930, 7189973830216081696, 1298090729995668204288, 215276329320562758744210, 33531967207612008887673350

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

nonn

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
V. Jovovic and 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)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051116, A051117, A051118.

A051116 Number of monotone Boolean functions of n variables with 8 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 115, 1613250, 4693213105, 5971431466764, 4657267944250425, 2654563364004395160, 1223795727111874798255, 485987045749653063943998, 173253367143529540187635315, 57037488183550191520963561230

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

nonn

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
V. Jovovic and 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)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051115, A051117, A051118.

A051117 Number of monotone Boolean functions of n variables with 9 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 20, 1484230, 15946757960, 60089234465176, 122281201867047920, 168329227672583040430, 178185327268349957044060, 156921594738520322214197672, 121014019160263331691800711500

Offset: 0

Vladeta Jovovic, Goran Kilibarda, and Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
V. Jovovic and 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)
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, Belgrade, 1999, in preparation.

K. S. Brown, Dedekind's Problem
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051112, A051113, A051114, A051115, A051116, A051118.

cp's OEIS Frontend

A094037 Number of connected 6-element antichains on a labeled n-set.

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A084883 Number of (k,m,n)-multiantichains of multisets with k=3 and m=6.

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A069403 a(n) = 2*Fibonacci(2*n+1) - 1.

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A025211 Expansion of 1/((1-2x)(1-3x)(1-4x)(1-5x)).

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A051113 Number of monotone Boolean functions of n variables with 5 mincuts.

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A051114 Number of monotone Boolean functions of n variables with 6 mincuts.

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A027650 Poly-Bernoulli numbers B_n^(k) with k=-3.

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Magma

Maple

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A051115 Number of monotone Boolean functions of n variables with 7 mincuts.

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A069403 a(n) = 2Fibonacci(2n+1) - 1.