A016269 -id:A016269 - cp's OEIS frontend

A069396 Half the number of 3 X n binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

1, 25, 377, 4541, 48329, 476389, 4461489, 40306317, 354713977, 3060942133, 26020259201, 218626028573, 1820140085705, 15043088032837, 123602247055953, 1010793162739629, 8234370308667673, 66870924588036181

Offset: 2

R. H. Hardin, Mar 22 2002

nonn

G. C. Greubel, Table of n, a(n) for n = 2..1000

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.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2-6*x+1))); // G. C. Greubel, Apr 22 2018

Drop[CoefficientList[Series[x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x + 1)/(4*x^2 - 6*x + 1), {x, 0, 50}], x], 2] (* G. C. Greubel, Apr 22 2018 *)

x='x+O('x^30); Vec(x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2 -6*x+1)) \\ G. C. Greubel, Apr 22 2018

G.f.: x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2-6*x+1). - Vladeta Jovovic, Jul 02 2003

2*a(n) = 8^n+A084326(n+1) -2*A186446(n). - R. J. Mathar, May 09 2023

A069416 Half the number of n X 16 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

32767, 2104469695, 123602247055953, 6475978445076745163

Offset: 1

R. H. Hardin, Mar 22 2002

nonn

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.

A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.

Original entry on oeis.org

0, 0, 0, 0, 25, 2020, 82115, 2401910, 58089465, 1245331920, 24625121455, 460316430970, 8266174350005, 144171200793620, 2461016066613195, 41343340015862430, 686274244801356145, 11289648429330100120

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Zoran Maksimovic

J. L. Arocha, Antichains in ordered sets, (in Spanish) An. Inst. Mat. UNAM, vol. 27, 1987, 1-21.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 293, #8, s(n,4).
V. Jovovic, G. Kilibarda, On enumeration of the class of all monotone Boolean function, Belgrade, 1999, in preparation.

Charles R Greathouse IV, Table of n, a(n) for n = 0..831 (next term has 1001 digits)
K. S. Brown, Dedekind's Problem
D. M. Cvetkovic, The number of antichains of finite power sets, Publ. Inst. Math., 13 (27), 1972, 5-9.
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 linear recurrences with constant coefficients, signature (82, -2970, 62700, -856713, 7947786, -51019100, 226259000, -678011136, 1304341632, -1445575680, 696729600).
Index entries for sequences related to Boolean functions

Cf. A016269, A047707, A051113, A051114, A051115, A051116, A051117, A051118.

Table[(1/4!)*(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n-22*3^n+6*2^n),{n,0,20}] (* or *) LinearRecurrence[{82,-2970,62700,-856713,7947786,-51019100,226259000,-678011136,1304341632,-1445575680,696729600},{0,0,0,0,25,2020,82115,2401910,58089465,1245331920,24625121455},20] (* Harvey P. Dale, Nov 26 2019 *)

a(n)=(16^n-12*12^n+24*10^n+4*9^n-18*8^n+6*7^n-36*6^n+36*5^n+11*4^n -22*3^n+6*2^n)/24 \\ Charles R Greathouse IV, Mar 14 2012

a(n) = (1/4!)*(16^n - 12*12^n + 24*10^n + 4*9^n - 18*8^n + 6*7^n - 36*6^n + 36*5^n + 11*4^n - 22*3^n + 6*2^n).
From Michael Somos: (Start)
a(n) = 82*a(n - 1) - 2970*a(n - 2) + 62700*a(n - 3) - 856713*a(n - 4) + 7947786*a(n - 5) - 51019100*a(n - 6) + 226259000*a(n - 7) - 678011136*a(n - 8) + 1304341632*a(n - 9) - 1445575680*a(n - 10) + 696729600*a(n - 11).
G.f.: 5x^4(5-6x-1855x^2+20076x^3-44356x^4-215280x^5+759168x^6) / ((1-3x)(1-4x)(1-5x)(1-6x)(1-2x)(1-7x)(1-8x)(1-9x)(1-10x)(1-12x)(1-16x)). (End)

A047707 Number of monotone Boolean functions of n variables with 3 mincuts. Also Sperner systems with 3 blocks.

Original entry on oeis.org

0, 0, 0, 2, 64, 1090, 14000, 153762, 1533504, 14356610, 128722000, 1119607522, 9528462944, 79817940930, 660876543600, 5424917141282, 44246078560384, 359144709794050, 2904688464582800, 23429048035827042, 188593339362097824

Offset: 0

N. J. A. Sloane

The paper by G. Kilibarda, Enumeration of certain classes of antichains, Publications de l'Institut Mathematique, Nouvelle série, 97 (111) (2015), mentions many sequences, but since only very condensed formulas are given, it is hard to match them with entries in the OEIS. It would be nice to add this reference to all the sequences that it mentions. - N. J. A. Sloane, Jan 01 2016

Term a(1108) has 1000 decimal digits. - Michael De Vlieger, Jan 26 2016

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,3).

Michael De Vlieger, Table of n, a(n) for n = 0..1107
K. S. Brown, Dedekind's problem.
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
G. Kilibarda, Enumeration of certain classes of antichains, Publications de l'Institut Mathematique, Nouvelle série, 97 (111) (2015), 69-87 DOI: 10.2298/PIM140406001K. See page 86, formula for alpha^hat(3,n).
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (28,-315,1820,-5684,9072,-5760).

Cf. A016269, A051112.

Table[Binomial[2^n, 3] - (6^n - 5^n - 4^n + 3^n), {n, 20}] (* or *)
CoefficientList[Series[-2 x^3 (36 x^2 - 4 x - 1)/((2 x - 1) (3 x - 1) (4 x - 1) (5 x - 1) (6 x - 1) (8 x - 1)), {x, 0, 20}], x] (* Michael De Vlieger, Jan 26 2016 *)

a(n)=binomial(2^n,3)-(6^n-5^n-4^n+3^n) \\ Charles R Greathouse IV, Apr 08 2016

a(n) = (2^n)*(2^n - 1)*(2^n - 2)/6 - (6^n - 5^n - 4^n + 3^n).
G.f.: -2*x^3*(36*x^2-4*x-1)/((2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Jul 31 2012
a(n) = Binomial(2^n,3) - (6^n - 5^n - 4^n + 3^n). - Ross La Haye, Jan 26 2016

A051118 Number of monotone Boolean functions of n variables with 10 mincuts.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1067771, 43506231489, 501425871595264, 2719674203584968630, 9172837864705015158979, 22524989249381408262409893, 44328073635887914351462953684, 74381256243136645820404637874910

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, A051116, A051117.

A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

Original entry on oeis.org

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

Offset: 2

Peter Bala, Aug 20 2008

This is the case r = 2 of the r-Stirling numbers of the second kind. The 2-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1 and 2 belong to distinct subsets.
More generally, the r-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the numbers 1, 2, ..., r belong to distinct subsets. The case r = 1 gives the usual Stirling numbers of the second kind A008277; for other cases see A143495 (r = 3) and A143496 (r = 4).
The lower unitriangular array of r-Stirling numbers of the second kind equals the matrix product P^(r-1) * S (with suitable offsets in the row and column indexing), where P is Pascal's triangle, A007318 and S is the array of Stirling numbers of the second kind, A008277.
For the definition of and entries relating to the corresponding r-Stirling numbers of the first kind see A143491. For entries on r-Lah numbers refer to A143497. The theory of r-Stirling numbers of both kinds is developed in [Broder].
From Peter Bala, Sep 19 2008: (Start)
Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-2)*E^n*x^2 = Sum_{k = 0..n} T(n+2,k+2)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k= 2..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_2(x) = x^2. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 2-Eulerian numbers E_2(n,j) := A144696(n,j): T(n,k) = 2!/k!*Sum_ {j = n-k..n-2} E_2(n,j)*binomial(j,n-k) for n >= k >= 2. (End)
From Wolfdieter Lang, Sep 29 2011: (Start)
T(n,k) = S(n,k,2), n>=k>=2, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column no. k from (A20) with k->2, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(2*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(2*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393. (End)

			Triangle begins
  n\k|...2....3....4....5....6....7
  =================================
  2..|...1
  3..|...2....1
  4..|...4....5....1
  5..|...8...19....9....1
  6..|..16...65...55...14....1
  7..|..32..211..285..125...20....1
  ...
T(4,3) = 5. The set {1,2,3,4} can be partitioned into three subsets such that 1 and 2 belong to different subsets in 5 ways: {{1}{2}{3,4}}, {{1}{3}{2,4}}, {{1}{4}{2,3}}, {{2}{3}{1,4}} and {{2}{4}{1,3}}; the remaining possibility {{1,2}{3}{4}} is not allowed.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    5   1       |  =  | 4   3   1      ||0  2   1       ||0  0  1       | ...
| 8   19   9   1   |     | 8   7   4  1   ||0  4   3  1    ||0  0  2  1    |
|16   65  55  14  1|     |16  15  11  6  1||0  8   7  4  1 ||0  0  4  3  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - x)). See A055248. (End)

Peter Bala, Factorising (r,b)-Stirling arrays
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 2.
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, 1982, Stanford University, Department of Computer Science.
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
C. B. Corcino, L. C. Hsu, and E. L. Tan, Asymptotic approximations of r-Stirling numbers, Approximation Theory Appl. 15, No. 3 13-25 (1999).
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Technical Report TR 99-05, July 1999, Universität Wien.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
M. d’Ocagne, Sur une classe de nombres remarquables, Amer. J. Math., Vol. 9 (1887), 353-380.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
Mark Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014.

A001047 (column 3), A005493 (row sums), A008277, A016269 (column 4), A025211 (column 5), A049444 (matrix inverse), A074051 (alt. row sums).

Cf. A055248, A143491, A143495, A143496, A143497.

with combinat: T := (n, k) -> (1/(k-2)!)*add ((-1)^(k-i)*binomial(k-2,i)*(i+2)^(n-2),i = 0..k-2): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do;

t[n_, k_] := StirlingS2[n, k] - StirlingS2[n-1, k]; Flatten[ Table[ t[n, k], {n, 2, 11}, {k, 2, n}]] (* Jean-François Alcover, Dec 02 2011 *)

@CachedFunction
def stirling2r(n, k, r) :
    if n < r: return 0
    if n == r: return 1 if k == r else 0
    return stirling2r(n-1,k-1,r) + k*stirling2r(n-1,k,r)
A143494 = lambda n,k: stirling2r(n, k, 2)
for n in (2..6):
    [A143494(n, k) for k in (2..n)] # Peter Luschny, Nov 19 2012

T(n+2,k+2) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+2)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - Stirling2(n-1,k) for n, k >= 2.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 2, with boundary conditions T(n,1) = T(1,n) = 0 for all n, T(2,2) = 1 and T(2,k) = 0 for k > 2. Special cases: T(n,2) = 2^(n-2); T(n,3) = 3^(n-2) - 2^(n-2).
As a sum of monomial functions of degree m: T(n+m,n) = Sum_{2 <= i_1 <= ... <= i_m <= n} (i_1*i_2*...*i_m). For example, T(6,4) = Sum_{2 <= i <= j <= 4} (i*j) = 2*2 + 2*3 + 2*4 + 3*3 + 3*4 + 4*4 = 55.
E.g.f. column k+2 (with offset 2): 1/k!*exp(2*x)*(exp(x) - 1)^k.
O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-2*x)*(1-3*x)*...*(1-k*x)).
E.g.f.: exp(2*t + x*(exp(t) - 1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+2,k+2) *x^k*t^n/n! = Sum_{n >= 0} B_n(2;x)*t^n/n! = 1 + (2 + x)*t/1! + (4 + 5*x + x^2)*t^2/2! + ..., where the row polynomial B_n(2;x) := Sum_{k = 0..n} T(n+2,k+2)*x^k denotes the 2-Bell polynomial.
Dobinski-type identities: Row polynomial B_n(2;x) = exp(-x)*Sum_{i >= 0} (i + 2)^n*x^i/i!. Sum_{k = 0..n} k!*T(n+2,k+2)*x^k = Sum_{i >= 0} (i + 2)^n*x^i/(1 + x)^(i+1).
The T(n,k) are the connection coefficients between falling factorials and the shifted monomials (x + 2)^(n-2). For example, from row 4 we have 4 + 5*x + x*(x - 1) = (x + 2)^2, while from row 5 we have 8 + 19*x + 9*x*(x - 1) + x*(x - 1)*(x - 2) = (x + 2)^3.
The row sums of the array are the 2-Bell numbers, B_n(2;1), equal to A005493(n-2). The alternating row sums are the complementary 2-Bell numbers, B_n(2;-1), equal to (-1)^n*A074051(n-2).
This array is the matrix product P * S, where P denotes the Pascal triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
Also, this array equals the transpose of the upper triangular array A126351. The inverse array is A049444, the signed 2-Stirling numbers of the first kind. See A143491 for the unsigned version of the inverse.
Let f(x) = exp(exp(x)). Then for n >= 1, the row polynomials R(n,x) are given by R(n+2,exp(x)) = 1/f(x)*(d/dx)^n(exp(2*x)*f(x)). Similar formulas hold for A008277, A039755, A105794, A111577 and A154537. - Peter Bala, Mar 01 2012

A000453 Stirling numbers of the second kind, S(n,4).

Original entry on oeis.org

1, 10, 65, 350, 1701, 7770, 34105, 145750, 611501, 2532530, 10391745, 42355950, 171798901, 694337290, 2798806985, 11259666950, 45232115901, 181509070050, 727778623825, 2916342574750, 11681056634501, 46771289738810, 187226356946265, 749329038535350

Offset: 4

N. J. A. Sloane

Given a set {1,2,3,4}, a(n) is the number of occurrences where the first 2 comes after the first '1', the first '3' after the first '2' and the first '4' after the first '3' in a list of n+3. For example, a(1): 1234; a(2): 11234, 12134, 12314, 12341, 12234, 12324, 12342, 12334, 12343, 12344. Related to the cereal box problem. - Kevin Nowaczyk, Aug 02 2007

a(n) is the number of partitions of [n] into 4 nonempty subsets. - Enrique Navarrete, Aug 27 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 4..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. Brandts and C. Cihangir, Counting triangles that share their vertices with the unit n-cube, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
M. Griffiths and I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 347
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Cf. A008277 (Stirling2 triangle), A016269, A056280 (Mobius transform).

A000453:=1/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

t={}; Do[f=StirlingS2[n, 4]; AppendTo[t, f], {n, 120}]; t (* Vladimir Joseph Stephan Orlovsky, Sep 27 2008 *)
CoefficientList[Series[1/((1 - x) (1 - 2 x) (1 - 3 x) (1 - 4 x)), {x, 0, 25}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 20 2011 *)
LinearRecurrence[{10, -35, 50, -24}, {1, 10, 65, 350}, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

a(n)=(4^n-4*3^n+6*2^n-4)/24 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)).
E.g.f.: (exp(x)-1)^4/4!.
a(n) = (4^n - 4*3^n + 6*2^n - 4)/24. - Kevin Nowaczyk, Aug 02 2007
a(n) = det(|s(i+4,j+3)|, 1 <= i,j <= n-4), where s(n,k) are Stirling numbers of the first kind. - Mircea Merca, Apr 06 2013
a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4). - Wesley Ivan Hurt, Oct 10 2021

A143395 Triangle read by rows: T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 9, 1, 0, 15, 55, 18, 1, 0, 31, 285, 205, 30, 1, 0, 63, 1351, 1890, 545, 45, 1, 0, 127, 6069, 15421, 7770, 1190, 63, 1, 0, 255, 26335, 116298, 95781, 24150, 2282, 84, 1, 0, 511, 111645, 830845, 1071630, 416451, 62370, 3990, 108, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

This is the Sheffer triangle (1,exp(x)*(exp(x)-1)) (Jobotinsky type). See the e.g.f. given by V. Jovovic below, and the W. Lang link under A006232 (second part) for general Sheffer remarks and the conversion to the umbral notation of S. Roman's book. - Wolfdieter Lang, Oct 08 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) counts the ways a set of size n can be partitioned into k nonempty blocks and then a nonempty subset chosen from each block. An example is given below.
This triangle is the particular case a = 1, b = 1, c = 0 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. A008277 is the case a = 1, b = 0, c = 0.
Define a polynomial sequence x_(n) by putting x_(0) = 1, x_(1) = x and for n >= 2 setting x_(n) = x*(x - (n+1))*(x - (n+2))*...*(x - (2*n-1)), that is, x_(n) = (-1)^(n+1)*n!*(x/(2*n - x))*binomial(2*n - x,n) for n >= 0. Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis polynomials x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k), n = 0,1,2,.... Examples are given below.
Matrix factorization: Let M be the infinite lower unit triangular array with (n,k)-th entry (2^(n+1-k)-1)*binomial(n,k). For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. It follows from the recurrence equation given in the Formula section that the infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle (but with the first row and column omitted). See the Example section. (End)
The Bell transform of 2^(n+1)-1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			T(3,2) = 9: {1}{2}<-3, {1}{3}<-2, {1}{2,3}, {2}{1}<-3, {2}{3}<-1, {2}{1,3}, {3}{1}<-2, {3}{2}<-1, {3}{1,2}.
Triangle begins:
  1;
  0,   1;
  0,   3,    1;
  0,   7,    9,     1;
  0,  15,   55,    18,    1;
  0,  31,  285,   205,   30,    1;
  0,  63, 1351,  1890,  545,   45,  1;
  0, 127, 6069, 15421, 7770, 1190, 63,  1;
  ...
From _Peter Bala_, Jan 07 2015: (Start)
T(4,2) = 55: There are 7 partitions of the set {1,2,3,4} into 2 blocks. For the 3 set partitions of the type {a,b}{c,d} we can choose a nonempty subset from each block in one of 3*3 ways giving 3*3*3 = 27 possibilities in all. The remaining 4 set partitions of {1,2,3,4} into 2 blocks are of the form {a,b,c}{d} and we can choose a nonempty subset from each block in 7*1 ways giving 4*7*1 = 28 possible choices. Thus in total T(4,2) = 27 + 28 = 55.
Recurrence equation example:
T(4,2) = sum {j = 1..3} (2^(4-j) - 1)*binomial(3,j)*T(j,1) = 7*3*1 + 3*3*3 + 1*1*7 = 55.
Connection constants:
Row 3 = [0, 7, 9, 1]. Hence x^3 = 7*x + 9*x*(x - 3) + x*(x - 4)*(x - 5); Row 4 = [0, 15, 55, 18, 1]. Hence x^4 = 15*x + 55*x*(x - 3) + 18*x*(x - 4)*(x - 5) + x*(x - 5)*(x - 6)*(x - 7).
With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1        \/1           \/1       \       / 1        \
| 3  1     ||0  1        ||0 1      |      | 3  1      |
| 7  6  1  ||0  3  1     ||0 0 1    |... = | 7  9  1   |
|15 21 9 1 ||0  7  6  1  ||0 0 3 1  |      |15 55 18 1 |
|...       ||0 15 21  9 1||0 0 7 6 1|      |...        |
|...       ||...         ||...      |      |           |
(End)

Alois P. Heinz, Rows n = 0..140, flattened
Eli Bagno, Riccardo Biagioli, and David Garber, Some identities involving second kind Stirling numbers of types B and D, arXiv:1901.07830 [math.CO], 2019.
Peter Bala, A 3 parameter family of generalized Stirling numbers
Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.
Index entries for sequences related to rooted trees

Columns k=0-9: A000007, A000225, A016269(n-2), A028025(n-3), A143399, A143400, A143401, A143402, A143403, A143404.
Diagonal: A000012.
T(2*n,n) gives A383869.
See also A048993, A008277, A007318, A143405 for row sums.

[[(&+[Binomial(n,j)*StirlingSecond(j,k)*k^(n-j): j in [k..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

T:= (n, k)-> add(binomial(n,t)*Stirling2(t,k)*k^(n-t), t=k..n):
seq(seq(T(n, k), k=0..n), n=0..11);

t[0, 0]=1; t[n_, k_]:= SeriesCoefficient[Exp[y*Exp[x]*(Exp[x]-1)], {x, 0, n}, {y, 0, k}]*n!; Table[t[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 05 2013, after Vladeta Jovovic *)
Table[If[n==k==0, 1, If[k==0, 0, Sum[Binomial[n, j]*StirlingS2[j, k]* k^(n-j), {j,k,n}]]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2019 *)

{T(n,k) = sum(j=k, n, binomial(n,j)*stirling(j,k,2)*k^(n-j))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 07 2019

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2^(n+1)-1, 10) # Peter Luschny, Jan 18 2016

G.f. for column k: x^k/Product_{t=k..2*k} (1-t*x).
T(n,k) = Sum_{t=k..n} C(n,t) * Stirling2(t,k) * k^(n-t).
E.g.f.: exp(y*exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
T(n,k) = Sum_{m=0..k} Stirling2(n,k+m)*(k+m)!/(m!*(k-m)!). - Vladimir Kruchinin, Apr 06 2011
Let P be Pascal's triangle A007318. The first column of the array exp(t*(P^2-P)) gives the row generating polynomials of this triangle.
The row polynomials R(n,t) satisfy the recurrence R(n+1,t) = t*(Sum_{k = 0..n} (2^(k+1)-1)*C(n,k)*R(n-k,t)) with R(0,t) = 1. For example, the row 4 polynomial R(4,t) = 15*t + 55*t^2 + 18*t^3 + t^4 = t*((7*t + 9*t^2 + t^3) + 3*3*(3*t+t^2) + 7*3*t + 15*1). - Peter Bala, Oct 12 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) = (1/k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(j + k)^n.
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} (2^(n-j+1) - 1)*binomial(n,j)*T(j,k) with T(0,0) = 1 and T(n,0) = 0 for n >= 1. This leads to the matrix factorization noted in the Comments section.
The inverse array is a signed version of A038455. (End)

A084869 Number of 2-multiantichains of an n-set.

Original entry on oeis.org

1, 2, 5, 17, 71, 317, 1415, 6197, 26591, 112157, 466775, 1923077, 7863311, 31972397, 129459335, 522571157, 2104535231, 8460991037, 33972711095, 136277478437, 546270602351, 2188566048077, 8764718254055, 35090241492917

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Jun 10 2003

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 x is not a subset of y and y is not 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 = y. - Ross La Haye, Jan 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, J. Integer Seqs., Vol. 7, 2004.
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.
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Cf. A000079, A000217, A000392, A016269, A032263, A034472, A047707, A051112-A051118, A084870-A084883.

Table[2^(2*n-1) - 3^n + 3*2^(n-1), {n, 0, 20}] (* Vaclav Kotesovec, Oct 30 2015 *)

a(n) = 2^(2*n-1)-3^n+3*2^(n-1); \\ Altug Alkan, Sep 12 2017

a(n) = (1/2!)*(4^n - 2*3^n + 3*2^n).
a(n) = 3*StirlingS2(n+1,4) + StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 11 2008
G.f.: -(13*x^2-7*x+1) / ((2*x-1)*(3*x-1)*(4*x-1)). - Colin Barker, Nov 27 2012
a(n) = 9*a(n-1) - 26*a(n-2) + 24*a(n-3). - Vaclav Kotesovec, Oct 30 2015
a(n) = 2^(2n-1) + 2^n + 2^(n-1) - 3^n = A000217(2^n+1) - A034472(n), for n >= 1. - Bob Selcoe, Sep 12 2017

A094033 Number of connected 2-element antichains on a labeled n-set.

Original entry on oeis.org

0, 0, 0, 3, 18, 75, 270, 903, 2898, 9075, 27990, 85503, 259578, 784875, 2366910, 7125303, 21425058, 64373475, 193317030, 580344303, 1741819338, 5227030875, 15684238350, 47059006503, 141189602418, 423593973075, 1270832250870

Offset: 0

Goran Kilibarda, Vladeta Jovovic, Apr 22 2004

nonn

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 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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Yahia Djemmada, Abdelghani Mehdaoui, László Németh, and László Szalay, The Fibonacci-Fubini and Lucas-Fubini numbers, arXiv:2407.04409 [math.CO], 2024. See p. 10.
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.
Adam Roman, Igor T. Podolak and Agnieszka Deszynska, On the number of clusterings in a hierarchical classification model with overlapping clusters, Schedae Informaticae, Volume 20, 2011.
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).

Cf. A016269, A047707.
Cf. A051112, A051113, A051114, A051115, A051116, A051117, A051118.
Cf. A094034, A094035, A094036, A094037.

[seq(stirling2(n,3)*3,n=0..26)]; # Zerinvary Lajos, Dec 06 2006

Table[3 StirlingS2[n, 3], {n, 0, 26}] (* Michael De Vlieger, Nov 30 2015 *)

x='x+O('x^50); concat([0,0,0],Vec(serlaplace((exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!))) \\ G. C. Greubel, Oct 06 2017

a(n) = 3 * A000392(n).
E.g.f.: (exp(3*x)-3*exp(2*x)+3*exp(x)-1)/2!.
From Colin Barker, Mar 31 2012: (Start)
a(n) = (3^n-3*2^n+3)/2.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: 3*x^3/((1-x)*(1-2*x)*(1-3*x)). (End)

cp's OEIS Frontend

A069396 Half the number of 3 X n binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

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A069416 Half the number of n X 16 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

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A051112 Number of monotone Boolean functions of n variables with 4 mincuts. Also Sperner systems with 4 blocks.

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A047707 Number of monotone Boolean functions of n variables with 3 mincuts. Also Sperner systems with 3 blocks.

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A051118 Number of monotone Boolean functions of n variables with 10 mincuts.

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A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

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A000453 Stirling numbers of the second kind, S(n,4).

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A143395 Triangle read by rows: T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0 <= k <= n.

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