A016269 -id:A016269 - cp's OEIS frontend

A019333 Expansion of g.f. 1/((1-4x)(1-6x)(1-8*x)).

1, 18, 220, 2280, 21616, 194208, 1685440, 14290560, 119232256, 983566848, 8047836160, 65462691840, 530198327296, 4280634482688, 34479631482880, 277245459333120, 2226418414452736, 17862092934217728, 143201285904793600, 1147437816702566400, 9190468809917464576

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (18,-104,192).

Equals 2^n * A016269.

Cf. A017389, A019943.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-4*x)*(1-6*x)*(1-8*x)))); // Vincenzo Librandi, Jul 02 2013

I:=[1, 18, 220]; [n le 3 select I[n] else 18*Self(n-1)-104*Self(n-2)+192*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Jul 02 2013

CoefficientList[Series[1 / ((1 - 4 x) (1 - 6 x) (1 - 8 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jul 02 2013 *)

Vec(1/((1-4*x)*(1-6*x)*(1-8*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2*4^n -9*6^n +8*8^n. - R. J. Mathar, Jun 29 2013
From Vincenzo Librandi, Jul 02 2013: (Start)
a(n) = 18*a(n-1) - 104*a(n-2) + 192*a(n-3) for n > 2.
a(n) = 14*a(n-1) - 48*a(n-2) + 4^n. (End)
E.g.f.: exp(4*x)*(2 - 9*exp(2*x) + 8*exp(4*x)). - Stefano Spezia, Jun 04 2024
From Seiichi Manyama, May 04 2025: (Start)
a(n) = Sum_{k=0..n} 2^k * 4^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2).
a(n) = Sum_{k=0..n} (-2)^k * 8^(n-k) * binomial(n+2,k+2) * Stirling2(k+2,2). (End)

A059119 Triangle a(n,m)=number of m-element antichains on a labeled n-set; number of monotone n-variable Boolean functions with m mincuts (lower units), m=0..binomial(n,floor(n,2)).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 1, 8, 9, 2, 1, 16, 55, 64, 25, 6, 1, 1, 32, 285, 1090, 2020, 2146, 1380, 490, 115, 20, 2, 1, 64, 1351, 14000, 82115, 304752, 759457, 1308270, 1613250, 1484230, 1067771, 635044, 326990, 147440, 57675, 19238, 5325, 1170, 190, 20, 1, 1

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 06 2001

Row sums give A000372.

			[1, 1],
[1, 2],
[1, 4, 1],
[1, 8, 9, 2],
[1, 16, 55, 64, 25, 6, 1],
[1, 32, 285, 1090, 2020, 2146, 1380, 490, 115, 20, 2], ...

V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean functions, in preparation.

Suresh Govindarajan and Deepak Aditya, Table of n, a(n) for n = 0..94
K. S. Brown, Dedekind's Problem
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).

Cf. A000372, A016269, A047707, A051112-A051118.

a(n, 0) = 1; a(n, 1) = 2^n; a(n, 2) = A016269(n); a(n, 3) = A047707(n); a(n, 4) = A051112(n); a(5, n) = A051113(n); a(6, n) = A051114(n); a(7, n) = A051115(n); a(8, n) = A051116(n); a(9, n) = A051117(n); a(10, n) = A051118(n).

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block.

			Matrix begins:
1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079
0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047
0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269
0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211
0, 0, 0,  0,  1,  20,  245, 2380, 20181, ...
0, 0, 0,  0,  0,   1,   27,  434,  5418, ...
0, 0, 0,  0,  0,   0,    1,   35,   714, ...
0, 0, 0,  0,  0,   0,    0,    1,    44, ...
0, 0, 0,  0,  0,   0,    0,    0,     1, ...
Triangle begins:
1;
1,  2;
1,  5,  4;
1,  9, 19,  8;
1, 14, 55, 65, 16;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A039810, A039814, A126350, A054654, A126353.

T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011

T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).

Original entry on oeis.org

1, 12, 97, 660, 4081, 23772, 133057, 724260, 3863761, 20308332, 105558817, 544039860, 2785713841, 14192221692, 72020501377, 364354427460, 1838822866321, 9262446387852, 46585947584737

Offset: 0

N. J. A. Sloane

As (0,0,1,12,97,...) this is the fourth binomial transform of cosh(x)-1. It is the binomial transform of A016269, when this has two leading zeros. Its e.g.f. is then exp(4x)cosh(x) - exp(4x). - Paul Barry, May 13 2003
This gives the third column of the Sheffer triangle A143495 (3-restricted Stirling2 numbers). See the e.g.f. below, and A193685 for comments on the general case. - Wolfdieter Lang, Oct 08 2011
From Kevin Long, Mar 25 2017: (Start)
In the power set poset 2^(n+2), a(n) gives the number of size 3 subposets {A,B,C} such that A subset of C, B subset of C, and A||B. By symmetry, it also counts the size 3 subposets {A,B,C} such that C subset of A, C subset of B, and A||B.
By the power set poset, I mean the subsets of [n+2] ordered by inclusion. A||B means A and B are incomparable.
The result can be proved by showing that the formula holds. 5^n counts triples (A,B,C) of subsets of [n] where A subset of  C and B subset of C, since for each x in [n], it is either in C only, in A and C, in B and C, in all three, or in none. However, this also counts the cases where A subset of B and where B subset of A, and we want A||B.
Each case can be counted by 4^n, since if A subset of B⊆C, then each element x of [n] is either in all three, in B and C, in only C, or in none. Hence we subtract 2*4^n from 5^n. These two cases intersect, however, when A = B subset of C, which can be counted by 3^n, since each element x of [n] can be either in all three sets, in only C, or in none.
For the purposes of inclusion-exclusion, we add these sets back in to get 5^n-2*4^n+3^n to count all triples (A,B,C) where A subset of C, B subset of C, and A||B. We want sets, not triples, so this double-counts the sets since interchanging A and B give the same set, so we divide this by 2. Hence the formula for a(n) counts these subposets for 2^(n+2).  (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-47,60).

Cf. A000244, A005061, A016753, A132495, A193685, A245019.

[(5^(n+2) - 2*4^(n+2) + 3^(n+2))/2: n in [0..30]]; // G. C. Greubel, Sep 15 2018

CoefficientList[ Series[ 1/((1 - 3x)(1 - 4x)(1 - 5x)), {x, 0, 25} ], x ]
LinearRecurrence[{12,-47,60}, {1, 12, 97}, 30] (* G. C. Greubel, Sep 15 2018 *)

Vec(1/((1-3*x)*(1-4*x)*(1-5*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 5^(n+2)/2 - 4^(n+2) + 3^(n+2)/2. - Paul Barry, May 13 2003
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,2,3), (n >= 2). - Milan Janjic, Apr 26 2009
a(n) = 9*a(n-1) - 20*a(n-2) + 3^n, n >= 2. - Vincenzo Librandi, Mar 20 2011
O.g.f.: 1/((1-3*x)*(1-4*x)*(1-5*x)).
E.g.f.: (d^2/dx^2) (exp(3*x)*((exp(x)-1)^2)/2!). - Wolfdieter Lang, Oct 08 2011
a(n) = A245019(n+2)/2. - Kevin Long, Mar 24 2017

A038721 k=2 column of A038719.

Original entry on oeis.org

2, 18, 110, 570, 2702, 12138, 52670, 223290, 931502, 3842058, 15718430, 63928410, 258885902, 1045076778, 4208939390, 16921719930, 67944897902, 272553908298, 1092539107550, 4377127901850, 17529428119502, 70180466208618, 280910134414910, 1124205363178170, 4498515962822702

Offset: 1

N. J. A. Sloane, May 02 2000

For n>=1, a(n) is equal to the number of functions f: {1,2,...,n+1}->{1,2,3,4} such that Im(f) contains 2 fixed elements. - Aleksandar M. Janjic and Milan Janjic, Feb 27 2007
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if x is not a subset of y and y is not a subset of x. Then a(n+1) = |R|. - Ross La Haye, Mar 19 2009
Number of ordered (n+1)-tuples of positive integers, whose minimum is 0 and maximum 3. - Ovidiu Bagdasar, Sep 19 2014
a(n-2) is the number of possible player-reduced binary games observed by each player in an n X 2 game assuming k < n - 1 players reverse their initially fixed individual strategies and the remaining n - k - 1 players will play as one, either maintaining their status quo strategies or jointly adopting an alternative strategy. - Ambrosio Valencia-Romero, Apr 11 2024

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
O. Bagdasar, On Some Functions Involving the lcm and gcd of Integer Tuples, Scientific publications of the state university of Novi Pazar, Ser. A: Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
K. S. Immink, Coding Schemes for Multi-Level Channels that are Intrinsically Resistant Against Unknown Gain and/or Offset Using Reference Symbols, Electronics Letters, Volume50, Issue1, January 2014, pp. 20-22.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
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.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
Ambrosio Valencia-Romero and P. T. Grogan, The strategy dynamics of collective systems: Underlying hindrances beyond two-actor coordination, PLOS ONE 19(4): e0301394 (S1 Appendix).
Index entries for sequences related to posets
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A016269, A038719, A038720.

import Data.List (transpose)
a038721 n = a038721_list !! (n-1)
a038721_list = (transpose a038719_tabl) !! 2
-- Reinhard Zumkeller, Jul 08 2012

Table[4^n-2*3^n+2^n,{n,2,30}] (* or *) LinearRecurrence[{9,-26,24},{2,18,110},30] (* Harvey P. Dale, Aug 16 2012 *)

a(n) = 4^(n+1) - 2*3^(n+1) + 2^(n+1).
a(1)=2, a(2)=18, a(3)=110, a(n)=9*a(n-1)-26*a(n-2)+24*a(n-3). - Harvey P. Dale, Aug 16 2012
G.f.: -2*x/((2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Nov 27 2012
E.g.f.: 2*exp(2*x)*(1 - 3*exp(x) + 2*exp(2*x)). - Stefano Spezia, Jun 04 2024
a(n) = 2 * A016269(n-1). - Alois P. Heinz, Jun 04 2024

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

A094036 Number of connected 5-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 6, 2005, 280971, 22795136, 1345702092, 65250058251, 2781911443317, 108660434574142, 3991349973006198, 140293749275697017, 4775521611056597583, 158758002632650598268, 5185922974307536588224

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

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

E.g.f.: (exp(31*x)-20*exp(23*x)+60*exp(19*x)+20*exp(17*x)
+5*exp(16*x)-105*exp(15*x)-120*exp(14*x)+150*exp(13*x)+180*exp(12*x)
-300*exp(11*x)-110*exp(10*x)+380*exp(9*x)+160*exp(8*x)-575*exp(7*x)
+570*exp(6*x)-186*exp(5*x)-975*exp(4*x)+1645*exp(3*x)-1030*exp(2*x)
+274*exp(x)-24)/5!.

A069404 Half the number of n X 4 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

7, 55, 377, 2427, 15253, 94847, 587031, 3625675, 22372413, 137993145, 850987067, 5247512077, 32357022035, 199515609775, 1230218484787, 7585536760417, 46772417567873, 288398549126971, 1778263916566525, 10964764644841043, 67608669872179151, 416874624972396255

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..250
Sean A. Irvine, Java program (github)

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.

More terms from Sean A. Irvine, Aug 18 2024

A069405 Half the number of n X 5 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

15, 285, 4541, 66579, 944157, 13182673, 182702967, 2522968803, 34777826197, 478971480223, 6593672923115, 90751017499077, 1248904863846397, 17186379018703213, 236498372271010941, 3254365880309197587, 44781833050605593997, 616220311429717310963

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..250

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.

More terms from Sean A. Irvine, Aug 19 2024

A094034 Number of connected 3-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 1, 38, 645, 7510, 71981, 617358, 4947685, 37972070, 283229661, 2072354878, 14964711125, 107078983830, 761312910541, 5388481567598, 38017703680965, 267622831854790, 1880882526962621, 13203901505935518, 92616363612417205

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (22,-190,820,-1849,2038,-840).

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

With[{nmax = 50}, CoefficientList[Series[(Exp[7*x] - 6*Exp[5*x] + 3*Exp[4*x] + 14*Exp[3*x] - 21*Exp[2*x] + 11*Exp[x] - 2)/3!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)
LinearRecurrence[{22,-190,820,-1849,2038,-840},{0,0,0,1,38,645,7510},30] (* Harvey P. Dale, Sep 20 2022 *)

x='x+O('x^50); concat([0,0,0], Vec(-x^3*(5*x+1)*(56*x^2-11*x-1)/( (x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)))) \\ G. C. Greubel, Oct 07 2017

E.g.f.: (exp(7*x) - 6*exp(5*x) + 3*exp(4*x) + 14*exp(3*x) - 21*exp(2*x) + 11*exp(x) -2)/3!.

G.f.: -x^3*(5*x+1)*(56*x^2-11*x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(7*x-1)). - Colin Barker, Nov 27 2012

A094035 Number of connected 4-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 20, 1655, 65305, 1794730, 40179930, 793030245, 14423331635, 248261291960, 4113063835540, 66327037011235, 1049050826515965, 16360528085273190, 252545239130514350, 3869090307434050625, 58948119057416280295, 894447719738683138420

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

G. C. Greubel, Table of n, a(n) for n = 0..845

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

With[{nmax = 50}, CoefficientList[Series[(Exp[15*x] - 12*Exp[11*x] + 24*Exp[9*x] - 14*Exp[7*x] + 27*Exp[6*x] - 60*Exp[5*x] - 24*Exp[4*x] + 155*Exp[3*x] - 141*Exp[2*x] + 50*Exp[x] - 6)/4!, {x, 0, nmax}], x] Range[0, nmax]!] (* G. C. Greubel, Oct 07 2017 *)

x='x+O('x^50); concat([0,0,0,0], Vec(serlaplace((exp(15*x) -12*exp(11*x) +24*exp(9*x) -14*exp(7*x) +27*exp(6*x) -60*exp(5*x) -24*exp(4*x) +155*exp(3*x) -141*exp(2*x) +50*exp(x) -6)/4!))) \\ G. C. Greubel, Oct 07 2017

concat(vector(4), Vec(5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)) + O(x^30))) \\ Colin Barker, Oct 13 2017

E.g.f.: (exp(15*x) - 12*exp(11*x) + 24*exp(9*x) - 14*exp(7*x) + 27*exp(6*x) - 60*exp(5*x) - 24*exp(4*x) + 155*exp(3*x) - 141*exp(2*x) + 50*exp(x) - 6)/4!.

G.f.: 5*x^4*(4+79*x-988*x^2-4414*x^3+52260*x^4-8721*x^5-374220*x^6) / ((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-5*x)*(1-6*x)*(1-7*x)*(1-9*x)*(1-11*x)*(1-15*x)). - Colin Barker, Oct 13 2017

cp's OEIS Frontend

A019333 Expansion of g.f. 1/((1-4*x)*(1-6*x)*(1-8*x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Magma

Mathematica

PARI

Formula

A059119 Triangle a(n,m)=number of m-element antichains on a labeled n-set; number of monotone n-variable Boolean functions with m mincuts (lower units), m=0..binomial(n,floor(n,2)).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A016753 Expansion of 1/((1-3*x)*(1-4*x)*(1-5*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A038721 k=2 column of A038719.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

A094036 Number of connected 5-element antichains on a labeled n-set.

Views

Author

Keywords

Crossrefs

Formula

A069404 Half the number of n X 4 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Views

Author

Keywords

Links

Crossrefs

Extensions

A069405 Half the number of n X 5 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Views

Author

Keywords

Links

Crossrefs

Extensions

A094034 Number of connected 3-element antichains on a labeled n-set.

A019333 Expansion of g.f. 1/((1-4x)(1-6x)(1-8*x)).

A016753 Expansion of 1/((1-3x)(1-4x)(1-5*x)).