A016031 -id:A016031 - cp's OEIS frontend

A006125 a(n) = 2^(n*(n-1)/2).

1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832, 36028797018963968, 73786976294838206464, 302231454903657293676544, 2475880078570760549798248448, 40564819207303340847894502572032, 1329227995784915872903807060280344576

Offset: 0

N. J. A. Sloane

Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.
Number of perfect matchings of order n Aztec diamond. [see Speyer]
Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger]
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
From James Propp: (Start)
a(n) is the number of ways to tile the region
           o-----o
           |.....|
        o--o.....o--o
        |...........|
     o--o...........o--o
     |.................|
  o--o.................o--o
  |.......................|
  |.......................|
  |.......................|
  o--o.................o--o
     |.................|
     o--o...........o--o
        |...........|
        o--o.....o--o
           |.....|
           o-----o
(top-to-bottom distance = 2n) with dominoes like either of
  o--o      o-----o
  |..|  or  |.....|
  |..|      o-----o
  |..|
  o--o
(End)
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre, Apr 21 2002
Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g., a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy, Nov 10 2002
The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
The number of symmetric binary relations on an (n-1)-element set. - Peter Kagey, Feb 13 2021
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125. The probability of having won before n+1 tails is A114604 / A006125. - Joshua Zucker, Dec 14 2005
a(n) = A126883(n-1)+1. - Zerinvary Lajos, Jun 12 2007
Equals right border of triangle A158474 (unsigned). - Gary W. Adamson, Mar 20 2009
a(n-1) is the number of simple labeled graphs on n nodes such that every node has even degree. - Geoffrey Critzer, Oct 21 2011
a(n+1) is the number of symmetric binary matrices of size n X n. - Nathan J. Russell, Aug 30 2014
Let T_n be the n X n matrix with T_n(i,j) = binomial(2i + j - 3, j-1); then det(T_n) = a(n). - Tony Foster III, Aug 30 2018
k^(n*(n-1)/2) is the determinant of n X n matrix T_(i,j) = binomial(k*i + j - 3, j-1), in this case k=2. - Tony Foster III, May 12 2019
Let B_n be the n+1 X n+1 matrix with B_n(i, j) = Sum_{m=max(0, j-i)..min(j, n-i)} (binomial(i, j-m) * binomial(n-i, m) * (-1)^m), 0<=i,j<=n. Then det B_n = a(n+1). Also, deleting the first row and any column from B_n results in a matrix with determinant a(n). The matrices B_n have the following property: B_n * [x^n, x^(n-1) * y, x^(n-2) * y^2, ..., y^n]^T = [(x-y)^n, (x-y)^(n-1) * (x+y), (x-y)^(n-2) * (x+y)^2, ..., (x+y)^n]^T. - Nicolas Nagel, Jul 02 2019
a(n) is the number of positive definite (-1,1)-matrices of size n X n. - Eric W. Weisstein, Jan 03 2021
a(n) is the number of binary relations on a labeled n-set that are both total and antisymmetric. - José E. Solsona, Feb 05 2023

			From _Gus Wiseman_, Feb 11 2021: (Start)
This sequence counts labeled graphs on n vertices. For example, the a(0) = 1 through a(2) = 8 graph edge sets are:
  {}  {}  {}    {}
          {12}  {12}
                {13}
                {23}
                {12,13}
                {12,23}
                {13,23}
                {12,13,23}
This sequence also counts labeled graphs with loops on n - 1 vertices. For example, the a(1) = 1 through a(3) = 8 edge sets are the following. A loop is represented as an edge with two equal vertices.
  {}  {}    {}
      {11}  {11}
            {12}
            {22}
            {11,12}
            {11,22}
            {12,22}
            {11,12,22}
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 547 (Fig. 9.7), 573.
G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).
J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..50
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - From _N. J. A. Sloane_, Oct 08 2012
Anders Björner and Richard P. Stanley, A combinatorial miscellany, 2010.
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Taylor Brysiewicz and Fulvio Gesmundo, The Degree of Stiefel Manifolds, arXiv:1909.10085 [math.AG], 2019.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Mihai Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.
Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97.
Thierry de la Rue and Élise Janvresse, Pavages aléatoires par touillage de dominos, Images des Mathématiques, CNRS, 2023. In French.
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part I, Journal of Algebraic Combinatorics 1-2, 111-132 (1992).
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part II, Journal of Algebraic Combinatorics 1-3, 219-234 (1992).
Sen-Peng Eu and Tung-Shan Fu, A simple proof of the Aztec diamond theorem, arXiv:math/0412041 [math.CO], 2004.
D. Grensing, I. Carlsen, and H.-Chr. Zapp, Some exact results for the dimer problem on plane lattices with non-standard boundaries, Phil. Mag. A, 41:5 (1980), 777-781.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
William Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoretical Computer Science, Vol. 319, No. 1-3 (2004), pp. 29-57, arXiv preprint, arXiv:math/0304090 [math.CO], 2003.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991
W. H. Mills, David P. Robbins, and Howard Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
James Propp, Enumeration of matchings: problems and progress, arXiv:math/9904150 [math.CO], 1999.
James Propp, Lessons I learned from Richard Stanley, arXiv preprint [math.CO], 2015.
James Propp and R. P. Stanley, Domino tilings with barriers, arXiv:math/9801067 [math.CO], 1998.
Steven S. Skiena, Generating graphs.
David E. Speyer, Perfect matchings and the octahedron recurrence, Journal of Algebraic Combinatorics, Vol. 25, No. 3 (2007), pp. 309-348, arXiv preprint, arXiv:math/0402452 [math.CO], 2004.
Jan Tóth and Ondřej Kuželka, Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat, arXiv:2404.12905 [cs.LO], 2024. See p. 24.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Labeled Graph.
Eric Weisstein's World of Mathematics, Symmetric Matrix.
Doron Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.
Index entries for sequences related to dominoes

Cf. A000568 for the unlabeled analog, A053763, A006253, A004003.
Cf. A001187 (connected labeled graphs).
Cf. A095340, A103904, A005329, A114604, A299998.
Cf. A158474. - Gary W. Adamson, Mar 20 2009
Cf. A136652 (log). - Paul D. Hanna, Dec 04 2009
The unlabeled version is A000088, or A002494 without isolated vertices.
The directed version is A002416.
The covering case is A006129.
The version for hypergraphs is A058891, or A016031 without singletons.
Row sums of A143543.
The case of connected edge set is A287689.
Cf. A000169, A000295, A036442, A059167, A100743, A136284, A327078.

[2^(n*(n-1) `div` 2) | n <- [0..20]] -- José E. Solsona, Feb 05 2023

[2^(n*(n-1) div 2): n in [0..20]]; // Vincenzo Librandi, Jul 03 2019

Join[{1}, 2^Accumulate[Range[0, 20]]] (* Harvey P. Dale, May 09 2013 *)
Table[2^(n (n - 1) / 2), {n, 0, 20}] (* Vincenzo Librandi, Jul 03 2019 *)
Table[2^Binomial[n, 2], {n, 0, 15}] (* Eric W. Weisstein, Jan 03 2021 *)
2^Binomial[Range[0, 15], 2] (* Eric W. Weisstein, Jan 03 2021 *)
Prepend[Table[Count[Tuples[{0, 1}, {n, n}], ?SymmetricMatrixQ], {n, 5}], 1] (* _Eric W. Weisstein, Jan 03 2021 *)
Prepend[Table[Count[Tuples[{-1, 1}, {n, n}], ?PositiveDefiniteMatrixQ], 1], {n, 4}] (* _Eric W. Weisstein, Jan 03 2021 *)

A006125(n):=2^(n*(n-1)/2)$ makelist(A006125(n), n, 0, 30); /* Martin Ettl, Oct 24 2012 */

a(n)=1<Charles R Greathouse IV, Jun 15 2011

def A006125(n): return 1<<(n*(n-1)>>1) # Chai Wah Wu, Nov 09 2023

Sequence is given by the Hankel transform of A001003 (Schroeder's numbers) = 1, 1, 3, 11, 45, 197, 903, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - Philippe Deléham, Mar 02 2004
a(n) = 2^floor(n^2/2)/2^floor(n/2). - Paul Barry, Oct 04 2004
G.f. satisfies: A(x) = 1 + x*A(2x). - Paul D. Hanna, Dec 04 2009
a(n) = 2 * a(n-1)^2 / a(n-2). - Michael Somos, Dec 30 2012
G.f.: G(0)/x - 1/x, where G(k) = 1 + 2^(k-1)*x/(1 - 1/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013
E.g.f. satisfies A'(x) = A(2x). - Geoffrey Critzer, Sep 07 2013
Sum_{n>=1} 1/a(n) = A299998. - Amiram Eldar, Oct 27 2020
a(n) = s_lambda(1,1,...,1) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022
a(n) = Product_{1 <= j <= i <= n-1}  (i + j)/(2*i - 2*j + 1). Cf. A007685.  - Peter Bala, Oct 25 2024

More terms from Vladeta Jovovic, Apr 09 2000

A000295 Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

Original entry on oeis.org

0, 0, 1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, 67108837, 134217700, 268435427, 536870882, 1073741793, 2147483616, 4294967263, 8589934558

Offset: 0

N. J. A. Sloane

There are 2 versions of Euler's triangle:
* A008292 Classic version of Euler's triangle used by Comtet (1974).
* A173018 Version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990).
Euler's triangle rows and columns indexing conventions:
* A008292 The rows and columns of the Eulerian triangle are both indexed starting from 1. (Classic version: used in the classic books by Riordan and Comtet.)
* A173018 The rows and columns of the Eulerian triangle are both indexed starting from 0. (Graham et al.)
Number of Dyck paths of semilength n having exactly one long ascent (i.e., ascent of length at least two). Example: a(4)=11 because among the 14 Dyck paths of semilength 4, the paths that do not have exactly one long ascent are UDUDUDUD (no long ascent), UUDDUUDD and UUDUUDDD (two long ascents). Here U=(1,1) and D=(1,-1). Also number of ordered trees with n edges having exactly one branch node (i.e., vertex of outdegree at least two). - Emeric Deutsch, Feb 22 2004
Number of permutations of {1,2,...,n} with exactly one descent (i.e., permutations (p(1),p(2),...,p(n)) such that #{i: p(i)>p(i+1)}=1). E.g., a(3)=4 because the permutations of {1,2,3} with one descent are 132, 213, 231 and 312.
a(n+1) is the convolution of nonnegative integers (A001477) and powers of two (A000079). - Graeme McRae, Jun 07 2006
Partial sum of main diagonal of A125127. - Jonathan Vos Post, Nov 22 2006
Number of partitions of an n-set having exactly one block of size > 1. Example: a(4)=11 because, if the partitioned set is {1,2,3,4}, then we have 1234, 123|4, 124|3, 134|2, 1|234, 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3 and 1|2|34. - Emeric Deutsch, Oct 28 2006
k divides a(k+1) for k in A014741. - Alexander Adamchuk, Nov 03 2006
(Number of permutations avoiding patterns 321, 2413, 3412, 21534) minus one. - Jean-Luc Baril, Nov 01 2007, Mar 21 2008
The chromatic invariant of the prism graph P_n for n >= 3. - Jonathan Vos Post, Aug 29 2008
Decimal integer corresponding to the result of XORing the binary representation of 2^n - 1 and the binary representation of n with leading zeros. This sequence and a few others are syntactically similar. For n > 0, let D(n) denote the decimal integer corresponding to the binary number having n consecutive 1's. Then D(n).OP.n represents the n-th term of a sequence when .OP. stands for a binary operator such as '+', '-', '*', 'quotentof', 'mod', 'choose'. We then get the various sequences A136556, A082495, A082482, A066524, A000295, A052944. Another syntactically similar sequence results when we take the n-th term as f(D(n)).OP.f(n). For example if f='factorial' and .OP.='/', we get (A136556)(A000295) ; if f='squaring' and .OP.='-', we get (A000295)(A052944). - K.V.Iyer, Mar 30 2009
Chromatic invariant of the prism graph Y_n.
Number of labelings of a full binary tree of height n-1, such that each path from root to any leaf contains each label from {1,2,...,n-1} exactly once. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009
Also number of nontrivial equivalence classes generated by the weak associative law X((YZ)T)=(X(YZ))T on words with n open and n closed parentheses. Also the number of join (resp. meet)-irreducible elements in the pruning-grafting lattice of binary trees with n leaves. - Jean Pallo, Jan 08 2010
Nonzero terms of this sequence can be found from the row sums of the third sub-triangle extracted from Pascal's triangle as indicated below by braces:
                     1;
                  1,    1;
              {1},   2,    1;
           {1,    3},   3,    1;
        {1,    4,    6},   4,    1;
     {1,    5,   10,   10},   5,    1;
  {1,    6,   15,   20,   15},   6,    1;
  ... - L. Edson Jeffery, Dec 28 2011
For integers a, b, denote by a<+>b the least c >= a, such that the Hamming distance D(a,c) = b (note that, generally speaking, a<+>b differs from b<+>a). Then for n >= 3, a(n) = n<+>n. This has a simple explanation: for n >= 3 in binary we have a(n) = (2^n-1)-n = "anti n". - Vladimir Shevelev, Feb 14 2012
a(n) is the number of binary sequences of length n having at least one pair 01. - Branko Curgus, May 23 2012
Nonzero terms are those integers k for which there exists a perfect (Hamming) error-correcting code. - L. Edson Jeffery, Nov 28 2012
a(n) is the number of length n binary words constructed in the following manner:  Select two positions in which to place the first two 0's of the word.  Fill in all (possibly none) of the positions before the second 0 with 1's and then complete the word with an arbitrary string of 0's or 1's. So a(n) = Sum_{k=2..n} (k-1)*2^(n-k). - Geoffrey Critzer, Dec 12 2013
Without first 0: a(n)/2^n equals Sum_{k=0..n} k/2^k. For example: a(5)=57, 57/32 = 0/1 + 1/2 + 2/4 + 3/8 + 4/16 + 5/32. - Bob Selcoe, Feb 25 2014
The first barycentric coordinate of the centroid of the first n rows of Pascal's triangle, assuming the numbers are weights, is A000295(n+1)/A000337(n). See attached figure. - César Eliud Lozada, Nov 14 2014
Starting (0, 1, 4, 11, ...), this is the binomial transform of (0, 1, 2, 2, 2, ...). - Gary W. Adamson, Jul 27 2015
Also the number of (non-null) connected induced subgraphs in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Aug 27 2017
a(n) is the number of swaps needed in the worst case to transform a binary tree with n full levels into a heap, using (bottom-up) heapify. - Rudy van Vliet, Sep 19 2017
The utility of large networks, particularly social networks, with n participants is given by the terms a(n) of this sequence. This assertion is known as Reed's Law, see the Wikipedia link. - Johannes W. Meijer, Jun 03 2019
a(n-1) is the number of subsets of {1..n} in which the largest element of the set exceeds by at least 2 the next largest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,2,5}, {1,3,5}, {2,3,5}, {1,2,3,5}. - Enrique Navarrete, Apr 08 2020
a(n-1) is also the number of subsets of {1..n} in which the second smallest element of the set exceeds by at least 2 the smallest element. For example, for n = 5, a(4) = 11 and the 11 sets are {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {3,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,4,5}, {1,3,4,5}. - Enrique Navarrete, Apr 09 2020
a(n+1) is the sum of the smallest elements of all subsets of {1..n}. For example, for n=3, a(4)=11; the subsets of {1,2,3} are {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}, and the sum of smallest elements is 11. - Enrique Navarrete, Aug 20 2020
Number of subsets of an n-set that have more than one element. - Eric M. Schmidt, Mar 13 2021
Number of individual bets in a "full cover" bet on n-1 horses, dogs, etc. in different races. Each horse, etc. can be bet on or not, giving 2^n bets.  But, by convention, singles (a bet on only one race) are not included, reducing the total number bets by n.  It is also impossible to bet on no horses at all, reducing the number of bets by another 1.  A full cover on 4 horses, dogs, etc. is therefore 6 doubles, 4 trebles and 1 four-horse etc. accumulator.  In British betting, such a bet on 4 horses etc. is a Yankee; on 5, a super-Yankee. - Paul Duckett, Nov 17 2021
From Enrique Navarrete, May 25 2022: (Start)
Number of binary sequences of length n with at least two 1's.
a(n-1) is the number of ways to choose an odd number of elements greater than or equal to 3 out of n elements.
a(n+1) is the number of ways to split [n] = {1,2,...,n} into two (possibly empty) complementary intervals {1,2,...,i} and {i+1,i+2,...,n} and then select a subset from the first interval (2^i choices, 0 <= i <= n), and one block/cell (i.e., subinterval) from the second interval (n-i choices, 0 <= i <= n).
(End)
Number of possible conjunctions in a system of n planets; for example, there can be 0 conjunctions with one planet, one with two planets, four with three planets (three pairs of planets plus one with all three) and so on. - Wendy Appleby, Jan 02 2023
Largest exponent m such that 2^m divides (2^n-1)!. - Franz Vrabec, Aug 18 2023
It seems that a(n-1) is the number of odd r with 0 < r < 2^n for which there exist u,v,w in the x-independent beginning of the Collatz trajectory of 2^n x + r with u+v = w+1, as detailed in the link "Collatz iteration and Euler numbers?". A better understanding of this might also give a formula for A374527. - Markus Sigg, Aug 02 2024
This sequence has a connection to consecutively halved positional voting (CHPV); see Mendenhall and Switkay. - Hal M. Switkay, Feb 25 2025
a(n) is the number of subsets of size 2 and more of an n-element set. Equivalently, a(n) is the number of (hyper)edges of size 2 and more in a complete hypergraph of n vertices. - Yigit Oktar, Apr 05 2025

			G.f. = x^2 + 4*x^3 + 11*x^4 + 26*x^5 + 57*x^6 + 120*x^7 + 247*x^8 + 502*x^9 + ...

O. Bottema, Problem #562, Nieuw Archief voor Wiskunde, 28 (1980) 115.
L. Comtet, "Permutations by Number of Rises; Eulerian Numbers." Section 6.5 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 51 and 240-246, 1974.
F. N. David and D. E. Barton, Combinatorial Chance. Hafner, NY, 1962, p. 151.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, p. 34.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.
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 = 0..500
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"
Dillan Agrawal, Selena Ge, Jate Greene, Tanya Khovanova, Dohun Kim, Rajarshi Mandal, Tanish Parida, Anirudh Pulugurtha, Gordon Redwine, Soham Samanta, and Albert Xu, Chip-Firing on Infinite k-ary Trees, arXiv:2501.06675 [math.CO], 2025. See pp. 9, 17, 18.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Jean-Luc Baril and J. M. Pallo, The pruning-grafting lattice of binary trees, Theoretical Computer Science, 409, 2008, 382-393.
Jean-Luc Baril and José L. Ramírez, Some distributions in increasing and flattened permutations, arXiv:2410.15434 [math.CO], 2024. See p. 7.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, Maximilien Gadouleau, James D. Mitchell, and Yann Peresse, Chains of subsemigroups, arXiv preprint arXiv:1501.06394 [math.GR], 2015. See Table 4.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Shelby Cox, Pratik Misra, and Pardis Semnani, Homaloidal Polynomials and Gaussian Models of Maximum Likelihood Degree One, arXiv:2402.06090 [math.AG], 2024.
Benjamin Hellouin de Menibus and Yvan Le Borgne, Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automaton, arXiv:1903.12622 [math.PR], 2019.
Karl Dilcher and Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1) = 1, arXiv:1909.11222 [math.NT], 2019.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, J Algebr Comb (2016) 44: 363. DOI: 10.1007/s10801-016-0673-y.
Pascal Floquet, Serge Domenech and Luc Pibouleau, Combinatorics of Sharp Separation System synthesis : Generating functions and Search Efficiency Criterion, Industrial Engineering and Chemistry Research, 33, pp. 440-443, 1994.
Pascal Floquet, Serge Domenech, Luc Pibouleau and Said Aly, Some Complements in Combinatorics of Sharp Separation System Synthesis, American Institute of Chemical Engineering Journal, 39(6), pp. 975-978, 1993.
E. T. Frankel, A calculus of figurate numbers and finite differences, American Mathematical Monthly, 57 (1950), 14-25. [Annotated scanned copy]
Joël Gay, Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups, Doctoral Thesis, Discrete Mathematics [cs.DM], Université Paris-Saclay, 2018.
R. K. Guy, Letter to N. J. A. Sloane.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See pp. 24-25.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 388.
Wayne A. Johnson, An Euler operator approach to Ehrhart series, arXiv:2303.16991 [math.CO], 2023.
Lucas Kang, Investigation of Rule 73 as Case Study of Class 4 Long-Distance Cellular Automata, arXiv preprint arXiv:1310.3311 [nlin.CG], 2013.
Oliver Kullmann and Xishun Zhao, Bounds for variables with few occurrences in conjunctive normal forms, arXiv preprint arXiv:1408.0629 [math.CO], 2014.
César Eliud Lozada, Centroids of Pascal triangles
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
Peter Charles Mendenhall and Hal M. Switkay, Consecutively Halved Positional Voting: A Special Case of Geometric Voting, Social Sciences vol. 12 no. 2 (2023), 47-59.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971
J. W. Moon, A problem on arcs without bypasses in tournaments, J. Combinatorial Theory Ser. B 21 (1976), no. 1, 71--75. MR0427129(55 #165).
Agustín Moreno Cañadas, Hernán Giraldo, and Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Nagatomo Nakamura, Pseudo-Normal Random Number Generation via the Eulerian Numbers, Josai Mathematical Monographs, vol 8, p 85-95, 2015.
Emily Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Ronald Orozco López, Solution of the Differential Equation y^(k)= e^(a*y), Special Values of Bell Polynomials and (k,a)-Autonomous Coefficients, Universidad de los Andes (Colombia 2021).
J. M. Pallo, Weak associativity and restricted rotation, Information Processing Letters, 109, 2009, 514-517.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
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
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Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A008292 (classic version of Euler's triangle used by Comtet (1974)).
Cf. A173018 (version of Euler's triangle used by Graham, Knuth and Patashnik in Concrete Math. (1990)).
Cf. A002662 (partial sums).
Partial sums of A000225.
Row sums of A014473 and of A143291.
Second column of triangles A112493 and A112500.
Sequences A125128 and A130103 are essentially the same.
Column k=1 of A124324.
Cf. A000079, A000108, A000217, A000325, A000975, A001511, A002663.
Cf. A002664, A007814, A008292, A008949, A014741, A016031, A028366.
Cf. A035039, A035040, A035041, A035042, A079583, A107907, A130321.
Cf. A130128, A130330, A131768, A131816.

a000295 n = 2^n - n - 1  -- Reinhard Zumkeller, Nov 25 2013

[2^n-n-1: n in [0..40]]; // Vincenzo Librandi, Jul 29 2015

[EulerianNumber(n, 1): n in [0..40]]; // G. C. Greubel, Oct 02 2024

[ seq(2^n-n-1, n=1..50) ];
A000295 := -z/(2*z-1)/(z-1)**2; # Simon Plouffe in his 1992 dissertation
# Grammar specification:
spec := [S, { B = Set(Z, 1 <= card), C = Sequence(B, 2 <= card), S = Prod(B, C) }, unlabeled]:
struct := n -> combstruct[count](spec, size = n+1);
seq(struct(n), n = 0..33); # Peter Luschny, Jul 22 2014

a[n_] = If[n==0, 0, n*(HypergeometricPFQ[{1, 1-n}, {2}, -1] - 1)];
Table[a[n], {n,0,40}] (* Olivier Gérard, Mar 29 2011 *)
LinearRecurrence[{4, -5, 2}, {0, 0, 1}, 40] (* Vincenzo Librandi, Jul 29 2015 *)
Table[2^n -n-1, {n,0,40}] (* Eric W. Weisstein, Nov 16 2017 *)

a(n)=2^n-n-1 \\ Charles R Greathouse IV, Jun 10 2011

[2^n -(n+1) for n in range(41)] # G. C. Greubel, Oct 02 2024

a(n) = 2^n - n - 1.
G.f.: x^2/((1-2*x)*(1-x)^2).
A107907(a(n+2)) = A000079(n+2). - Reinhard Zumkeller, May 28 2005
E.g.f.: exp(x)*(exp(x)-1-x). - Emeric Deutsch, Oct 28 2006
a(0)=0, a(1)=0, a(n) = 3*a(n-1) - 2*a(n-2) + 1. - Miklos Kristof, Mar 09 2005
a(0)=0, a(n) = 2*a(n-1) + n - 1 for all n in Z.
a(n) = Sum_{k=2..n} binomial(n, k). - Paul Barry, Jun 05 2003
a(n+1) = Sum_{i=1..n} Sum_{j=1..i} C(i, j). - Benoit Cloitre, Sep 07 2003
a(n+1) = 2^n*Sum_{k=0..n} k/2^k. - Benoit Cloitre, Oct 26 2003
a(0)=0, a(1)=0, a(n) = Sum_{i=0..n-1} i+a(i) for i > 1. - Gerald McGarvey, Jun 12 2004
a(n+1) = Sum_{k=0..n} (n-k)*2^k. - Paul Barry, Jul 29 2004
a(n) = Sum_{k=0..n} binomial(n, k+2); a(n+2) = Sum_{k=0..n} binomial(n+2, k+2). - Paul Barry, Aug 23 2004
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k+1)*2^(n-k-2)*(-1/2)^k. - Paul Barry, Oct 25 2004
a(0) = 0; a(n) = Stirling2(n,2) + a(n-1) = A000225(n-1) + a(n-1). - Thomas Wieder, Feb 18 2007
a(n) = A000325(n) - 1. - Jonathan Vos Post, Aug 29 2008
a(0) = 0, a(n) = Sum_{k=0..n-1} 2^k - 1. - Doug Bell, Jan 19 2009
a(n) = A000217(n-1) + A002662(n) for n>0. - Geoffrey Critzer, Feb 11 2009
a(n) = A000225(n) - n. - Zerinvary Lajos, May 29 2009
a(n) = n*(2F1([1,1-n],[2],-1) - 1). - Olivier Gérard, Mar 29 2011
Column k=1 of A173018 starts a'(n) = 0, 1, 4, 11, ... and has the hypergeometric representation n*hypergeom([1, -n+1], [-n], 2). This can be seen as a formal argument to prefer Euler's A173018 over A008292. - Peter Luschny, Sep 19 2014
E.g.f.: exp(x)*(exp(x)-1-x); this is U(0) where U(k) = 1 - x/(2^k - 2^k/(x + 1 - x^2*2^(k+1)/(x*2^(k+1) - (k+1)/U(k+1)))); (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Dec 01 2012
a(n) = A079583(n) - A000225(n+1). - Miquel Cerda, Dec 25 2016
a(0) = 0; a(1) = 0; for n > 1: a(n) = Sum_{i=1..2^(n-1)-1} A001511(i). - David Siegers, Feb 26 2019
a(n) = A007814(A028366(n)). - Franz Vrabec, Aug 18 2023
a(n) = Sum_{k=1..floor((n+1)/2)} binomial(n+1, 2*k+1). - Taras Goy, Jan 02 2025

A323818 Number of connected set-systems covering n vertices.

Original entry on oeis.org

1, 1, 4, 96, 31840, 2147156736, 9223372011084915712, 170141183460469231602560095199828453376, 57896044618658097711785492504343953923912733397452774312021795134847892828160

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.

			The a(2) = 4 set-systems:
  {{1, 2}}
  {{1}, {1,2}}
  {{2}, {1,2}}
  {{1}, {2}, {1,2}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A001187, A003465 (not necessarily connected), A016031, A048143, A092918, A293510, A317672, A323816, A323817 (no singletons), A323819 (unlabeled case).

m:=12;
f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022

b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

nn=8;
ser=Sum[2^(2^n-1)*x^n/n!,{n,0,nn}];
Table[SeriesCoefficient[1-x+Log[ser],{x,0,n}]*n!,{n,0,nn}]

m=12;
def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))
def A_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A_list(m) # G. C. Greubel, Oct 04 2022

E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).

Logarithmic transform of A003465.

A367867 Number of labeled simple graphs with n vertices contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 416, 24244, 1951352, 265517333, 68652859502, 35182667175398, 36028748718835272, 73786974794973865449, 302231454853009287213496, 2475880078568912926825399800, 40564819207303268441662426947840, 1329227995784915869870199216532048487

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

In the connected case, these are just graphs with more than one cycle.

			Non-isomorphic representatives of the a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The complement is A133686, connected A129271, covering A367869.
The connected case is A140638 (graphs with more than one cycle).
The covering case is A367868.
For set-systems we have A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A326754, A355739, A355740, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006125(n) - A133686(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367863 Number of n-vertex labeled simple graphs with n edges and no isolated vertices.

Original entry on oeis.org

1, 0, 0, 1, 15, 222, 3760, 73755, 1657845, 42143500, 1197163134, 37613828070, 1295741321875, 48577055308320, 1969293264235635, 85852853154670693, 4005625283891276535, 199166987259400191480, 10513996906985414443720, 587316057411626070658200, 34612299496604684775762261

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

			Non-isomorphic representatives of the a(4) = 15 graphs:
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A057500, unlabeled A001429.
The unlabeled version is A006649.
The non-covering version is A116508.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A003465, A006126, A305000, A316983, A319559, A323817, A326754, A367769, A367901, A367902, A367903.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[#]==n&]],{n,0,5}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform is A367862.

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n). - Andrew Howroyd, Dec 29 2023

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

A302545 Number of non-isomorphic multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 12, 23, 84, 204, 682, 1977, 6546, 21003, 72038, 248055, 888771, 3240578, 12152775, 46527471, 182339441, 729405164, 2979121279, 12407308136, 52670355242, 227725915268, 1002285274515, 4487915293698, 20434064295155, 94559526596293, 444527730210294, 2122005930659752

Offset: 0

Gus Wiseman, Jun 20 2018

nonn

A multiset partition is a finite multiset of finite nonempty multisets of positive integers. A singleton is a multiset of size 1. The weight of a multiset partition is the sum of sizes of its elements. Weight is generally not the same as number of vertices.
Also non-isomorphic multiset partitions of weight n with no endpoints, where an endpoint is a vertex appearing only once (degree 1). For example, non-isomorphic representations of the a(4) = 12 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}

			The a(4) = 12 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The set-system version is A330054 (no endpoints) or A306005 (no singletons).
Non-isomorphic multiset partitions are A007716.
Set-systems with no singletons are A016031.
Cf. A049311, A283877, A293606, A293607, A306008, A317533, A317794, A317795, A320665, A330053, A330055, A330058.

\\ compare with similar program for A007716.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j]*x^t)) + O(x*x^k), -k)}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 15 2023

Extended by Gus Wiseman, Dec 09 2019

Terms a(11) and beyond from Andrew Howroyd, Jan 15 2023

A367869 Number of labeled simple graphs covering n vertices and satisfying a strict version of the axiom of choice.

Original entry on oeis.org

1, 0, 1, 4, 34, 387, 5596, 97149, 1959938, 44956945, 1154208544, 32772977715, 1019467710328, 34473686833527, 1259038828370402, 49388615245426933, 2070991708598960524, 92445181295983865757, 4376733266230674345874, 219058079619119072854095, 11556990682657196214302036

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

Number of labeled n-node graphs with at most one cycle in each component and no isolated vertices. - Andrew Howroyd, Dec 30 2023

			The a(3) = 4 graphs:
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A129271.
The non-covering case is A133686, complement A367867.
The complement is A367868, connected A140638 (unlabeled A140636).
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A355740, A367770, A367863, A367902.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,5}]

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(sqrt(1/(1-t))*exp(t/2 - 3*t^2/4 - x)))} \\ Andrew Howroyd, Dec 30 2023

E.g.f.: exp(B(x) - x - 1) where B(x) is the e.g.f. of A129271. - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367868 Number of labeled simple graphs covering n vertices and contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 7, 381, 21853, 1790135, 250562543, 66331467215, 34507857686001, 35645472109753873, 73356936892660012513, 301275024409580265134121, 2471655539736293803311467943, 40527712706903494712385171632959, 1328579255614092966328511889576785109

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(4) = 7 graphs:
  {{1,2},{1,3},{1,4},{2,3},{2,4}}
  {{1,2},{1,3},{1,4},{2,3},{3,4}}
  {{1,2},{1,3},{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The connected case is A140638, unlabeled A140636.
The non-covering case is A367867.
The complement is A367869, connected A129271, non-covering A133686.
The version for set-systems is A367903, ranks A367907.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A143543 counts simple labeled graphs by number of connected components.
Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) = A006129(n) - A367869(n). - Andrew Howroyd, Dec 30 2023

Terms a(7) and beyond from Andrew Howroyd, Dec 30 2023

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

Original entry on oeis.org

1, 1, 1, 2, 20, 308, 5338, 105298, 2366704, 60065072, 1702900574, 53400243419, 1836274300504, 68730359299960, 2782263907231153, 121137565273808792, 5645321914669112342, 280401845830658755142, 14788386825536445299398, 825378055206721558026931, 48604149005046792753887416

Offset: 0

Gus Wiseman, Dec 07 2023

nonn

Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

			Non-isomorphic representatives of the a(4) = 20 graphs:
  {}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,4},{2,3}}
  {{1,2},{1,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The connected case is A057500, unlabeled A001429.
Counting all vertices (not just covered) gives A116508.
The covering case is A367863, unlabeled A006649.
For set-systems we have A367916, ranks A367917.
A001187 counts connected graphs, A001349 unlabeled.
A003465 counts covering set-systems, unlabeled A055621, ranks A326754.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

\\ Here b(n) is A367863(n)
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A367863.

Terms a(8) and beyond from Andrew Howroyd, Dec 29 2023

A136556 a(n) = binomial(2^n - 1, n).

Original entry on oeis.org

1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, 396861704798625, 6098989894499557055, 331001552386330913728641, 64483955378425999076128999167, 45677647585984911164223317311276545, 118839819203635450208125966070067352769535, 1144686912178270649701033287538093722740144666625

Offset: 0

Paul D. Hanna, Jan 07 2008; Paul Hanna and Vladeta Jovovic, Jan 15 2008

nonn

Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.
Row 0 of square array A136555.
From Gus Wiseman, Dec 19 2023: (Start)
Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:
  {1}{2}{3}  {1}{2}{12}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{13}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{2}{23}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{12}  {1}{12}{23}  {2}{12}{123}
             {1}{3}{13}  {1}{13}{23}  {2}{13}{123}
             {1}{3}{23}  {2}{3}{123}  {2}{23}{123}
             {2}{3}{12}  {2}{12}{13}  {3}{12}{123}
             {2}{3}{13}  {2}{12}{23}  {3}{13}{123}
             {2}{3}{23}  {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...

G. C. Greubel, Table of n, a(n) for n = 0..50

Sequences of the form binomial(2^n +p*n +q, n): this sequence (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A066384, A136555, A136557.
The covering case A054780 has binomial transform A367916, ranks A367917.
Connected graphs of this type are A057500, unlabeled A001429.
Graphs of this type are A116508, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, connected A323818, without singletons A016031.
Cf. A055154, A059201, A133686, A367862, A367902, A367903.

[Binomial(2^n -1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A136556:= n-> binomial(2^n-1,n); seq(A136556(n), n=0..20); # G. C. Greubel, Mar 14 2021

f[n_] := Binomial[2^n - 1, n]; Array[f, 12] (* Robert G. Wilson v *)
Table[Length[Subsets[Rest[Subsets[Range[n]]],{n}]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

{a(n) = binomial(2^n-1,n)}
for(n=0, 20, print1(a(n), ", "))

/* As coefficient of x^n in the g.f.: */
{a(n) = polcoeff( sum(i=0,n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A136556(n): return comb((1<Chai Wah Wu, Jan 02 2024

[binomial(2^n -1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2^n,k).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.
G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).
a(n) ~ 2^(n^2)/n!. - Vaclav Kotesovec, Jul 02 2016

Edited by N. J. A. Sloane, Jan 26 2008

cp's OEIS Frontend

A006125 a(n) = 2^(n*(n-1)/2).

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