A038721 -id:A038721 - cp's OEIS frontend

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

1, 9, 55, 285, 1351, 6069, 26335, 111645, 465751, 1921029, 7859215, 31964205, 129442951, 522538389, 2104469695, 8460859965, 33972448951, 136276954149, 546269553775, 2188563950925, 8764714059751, 35090233104309, 140455067207455, 562102681589085, 2249257981411351

Offset: 0

N. J. A. Sloane

Half the number of 2 X (n+2) binary arrays with both a path of adjacent 1's and a path of adjacent 0's from top row to bottom row. - R. H. Hardin, Mar 21 2002
As (0,0,1,9,55,...) this is the third binomial transform of cosh(x)-1. It is the binomial transform of A000392, when this has two leading zeros. Its e.g.f. is then exp(3x)cosh(x) - exp(3x) and a(n) = (4^n - 2*3^n + 2^n)/2. - Paul Barry, May 13 2003
Let P(A) be the power set of an n-element set A. Then a(n-2) is 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. - Ross La Haye, Jan 10 2008
a(n) also gives the third column sequence 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
a(n) is also the number of even binomial coefficients in rows 0 through 2^(n+1)-1 of Pascal's triangle. - Aaron Meyerowitz, Oct 29 2013

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 292, #8, s(n,2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. S. Brown, Dedekind's problem
John Elias, Illustration of Initial Terms: Inverse of the Sierpinski Triangle
Vladeta Jovovic, Illustration for A016269, A047707, A051112-A051118
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.
N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92). - _N. J. A. Sloane_, May 12 2012
Index entries for sequences related to Boolean functions
Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Equals (1/2) A038721(n+1). First differences of A000453. Partial sums of A027650. Pairwise sums of A099110. Odd part of A019333.

Cf. A000079, A000392, A001047, A006516, A032263, A143494, A193685.

[(2^n)*(2^n-1)/2-3^n+2^n: n in [2..30]]; // Vincenzo Librandi, Oct 06 2017

a:= n-> Stirling2(n+4, 4)-Stirling2(n+3, 4): seq(a(n), n=0..24); # Zerinvary Lajos, Oct 05 2007

CoefficientList[1/((1-2x)(1-3x)(1-4x)) + O[x]^30, x] (* Jean-François Alcover, Nov 28 2015 *)
LinearRecurrence[{9, -26, 24}, {1, 9, 55}, 40] (* Vincenzo Librandi, Oct 06 2017 *)

a(n)=(2^n)*(2^n-1)/2-3^n+2^n \\ Charles R Greathouse IV, Mar 22 2016

G.f.: 1/((1-2*x)*(1-3*x)*(1-4*x)).
a(n-2) = (2^n)*(2^n - 1)/2 - 3^n + 2^n.
From Hieronymus Fischer, Jun 25 2007: (Start)
a(n) = Sum_{0<=i,j,k,<=n, i+j+k=n} 2^i*3^j*4^k.
a(n) = 2^(n+1)*(1+2^(n+2))-3^(n+2). (End)
a(n) = 3*StirlingS2(n+3,4) + StirlingS2(n+3,3). - Ross La Haye, Jan 10 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,2,2), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2) (exp(2*x)*((exp(x)-1)^2)/2!). See the Sheffer comment given above. - Wolfdieter Lang, Oct 08 2011
a(n) = A006516(n+2) - A001047(n+2). - Ross La Haye, Jan 26 2016
a(n) = A006516(n+1) + 3*a(n-1), n>=1, a(0)=1. - Carlos A. Rico A., Jun 22 2019

A038720 a(n) = (n+3)*n!/2.

Original entry on oeis.org

2, 5, 18, 84, 480, 3240, 25200, 221760, 2177280, 23587200, 279417600, 3592512000, 49816166400, 741015475200, 11769069312000, 198766503936000, 3556874280960000, 67224923910144000, 1338096104497152000, 27978373094031360000, 613091306060513280000

Offset: 1

N. J. A. Sloane, May 02 2000

Next-to-last diagonal of A038719.

a(n-1) is the sum of the n-th entries in all cycles of all permutations of [n]. a(2) = 5 because the sum of the third entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 3+2+0+0+0+0 = 5. - Alois P. Heinz, May 03 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to posets.

Cf. A038719, A038721, A052572, A214178.

Main diagonal of A285793.

import Data.List (transpose)
a038720 n = a038720_list !! (n-1)
a038720_list = (transpose $ map reverse a038719_tabl) !! 1
-- Reinhard Zumkeller, Jul 08 2012

A038720:= func< n | (n+3)*Factorial(n)/2 >; // G. C. Greubel, May 11 2025

Array[(# + 3) #!/2 &, 21] (* Michael De Vlieger, Apr 28 2022 *)

def A038720(n): return (n+3)*factorial(n)//2 # G. C. Greubel, May 11 2025

a(n) = A052572(n)/2.
a(n) = A214178(n+3,n). - Reinhard Zumkeller, Jul 08 2012
G.f.: Sum_{n>=1} ( (n+1)*x/(1 + (n+1)*x) )^n. - Paul D. Hanna, Jan 02 2013
E.g.f.: 1/(1-x) + 1/(2*(x-1)^2) - 3/2. - Alois P. Heinz, May 04 2017
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*e - 14/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10/e - 10/3. (End)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 09 2000.

A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 8, 19, 18, 6, 16, 65, 110, 84, 24, 32, 211, 570, 750, 480, 120, 64, 665, 2702, 5460, 5880, 3240, 720, 128, 2059, 12138, 35406, 57120, 52080, 25200, 5040, 256, 6305, 52670, 213444, 484344, 650160, 514080, 221760, 40320, 512, 19171

Offset: 0

N. J. A. Sloane, May 02 2000

The relation of this triangle to A143494 given in the Formula section leads to the following combinatorial interpretation: T(n,k) gives the number of partitions of the set {1,2,...,n+2} into k + 2 blocks where 1 and 2 belong to two distinct blocks and the remaining k blocks are labeled from a fixed set of k labels. - Peter Bala, Jul 10 2014

Also, the number of distinct k-level fuzzy subsets of a set consisting of n elements ordered by set inclusion. - Rajesh Kumar Mohapatra, Mar 16 2020

			Triangle begins
   1;
   2,   1;
   4,   5,   2;
   8,  19,  18,   6;
  16,  65, 110,  84,  24;
  ...
From _Peter Bala_, Feb 02 2022: (Start)
Table of successive differences of k^2 starting at k = 2
4   9   16
  5   7
    2
gives [4, 5, 2] as row 2 of this triangle.
Table of successive differences of k^3 starting at k = 2
8   27   64   125
  19   37   61
     18   24
        6
gives [8, 19, 8, 6] as row 3 of this triangle. (End)

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Deformations of the Hadamard product of power series
L. Bartlomiejczyk and J. Drewniak, A characterization of sets and operations invariant under bijections, Aequationes Mathematicae 68 (2004), pp. 1-9.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
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.
Index entries for sequences related to posets

Row sums give A007047. Columns give A000079, A001047, A038721. Next-to-last diagonal gives A038720.
Diagonal gives A000142. - Rajesh Kumar Mohapatra, Mar 16 2020
Cf. A028246. A019538, A143494.

a038719 n k = a038719_tabl !! n !! k
a038719_row n = a038719_tabl !! n
a038719_tabl = iterate f [1] where
   f row = zipWith (+) (zipWith (*) [0..] $ [0] ++ row)
                       (zipWith (*) [2..] $ row ++ [0])
-- Reinhard Zumkeller, Jul 08 2012

T:= proc(n, k) option remember;
      `if` (n=0, `if`(k=0, 1, 0), k*T(n-1, k-1) +(k+2)*T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 02 2011

t[n_, k_] := Sum[ (-1)^(k-i)*Binomial[k, i]*(2+i)^n, {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, after Pari *)

T(n,k)=sum(i=0,k,(-1)^(k-i)*binomial(k,i)*(2+i)^n)

T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k+2-j)^n.
T(n+1, k) = k*T(n, k-1) + (k+2)*T(n, k), T(0,0) = 1, T(0,k) = 0 for k>0.
E.g.f.: exp(2*x)/(1+y*(1-exp(x))). - Vladeta Jovovic, Jul 21 2003
A038719 as a lower triangular matrix is the binomial transform of A028246. - Gary W. Adamson, May 15 2005
Binomial transform of n-th row = 2^n + 3^n + 4^n + ...; e.g., binomial transform of [8, 19, 18, 6] = 2^3 + 3^3 + 4^3 + 5^3 + ... = 8, 27, 64, 125, ... - Gary W. Adamson, May 15 2005
From Peter Bala, Jul 09 2014: (Start)
T(n,k) = k!*( Stirling2(n+2,k+2) - Stirling2(n+1,k+2) ).
T(n,k) = k!*A143494(n+2,k+2).
n-th row polynomial = 1/(1 + x)*( sum {k >= 0} (k + 2)^n*(x/(1 + x))^k ). Cf. A028246. (End)
The row polynomials have the form (2 + x) o (2 + x) o ... o (2 + x), where o denotes the black diamond multiplication operator of Dukes and White. See example E12 in the Bala link. - Peter Bala, Jan 18 2018
Z(P,m) = Sum_{k=0..n} T(n,k)Binomial(m-2,k) = m^n, the zeta polynomial of the poset B_n. Each length m multichain from 0 to 1 in B_n corresponds to a function from [n] into [m]. - Geoffrey Critzer, Dec 25 2020
The entries in row n are the first terms in a table of the successive differences of the sequence [2^n, 3^n, 4^n, ...]. Examples are given below. - Peter Bala, Feb 02 2022

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

cp's OEIS Frontend

A016269 Number of monotone Boolean functions of n variables with 2 mincuts. Also number of Sperner systems with 2 blocks.

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A038720 a(n) = (n+3)*n!/2.

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A038719 Triangle T(n,k) (0 <= k <= n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion.

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Author

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Comments

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Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions