A095340 -id:A095340 - 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

A103904 a(n) = n(n-1)/2 2^(n*(n-1)/2).

Original entry on oeis.org

0, 2, 24, 384, 10240, 491520, 44040192, 7516192768, 2473901162496, 1583296743997440, 1981583836043018240, 4869940435459321626624, 23574053482485268906770432, 225305087149939210031640608768

Offset: 1

Ralf Stephan, Feb 21 2005

nonn

a(n) is the number of birooted graphs on n labeled nodes. - Andrew Howroyd, Nov 23 2020

Old (incorrect) name was: "Number of perfect matchings of an n X (n+1) Aztec rectangle with the third vertex in the topmost row removed". See Mathematics Stack Exchange for the discussion. - Andrey Zabolotskiy, Jun 05 2022

M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97, doi:10.1006/jcta.1996.2725.
N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings, Journal of Algebraic Combinatorics 1 (1992), 111-132 (Part I), 219-234 (Part II); arXiv:math/9201305 [math.CO], 1992.
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
C. Krattenthaler, Schur function identities and the number of perfect matchings of Aztec holey rectangles, arXiv:math/9712204 [math.CO], 1997.
Mathematics Stack Exchange, Mistake in OEIS A103904?, 2021.

Cf. A000217, A006125, A038094, A095340, A095351, A134401, A036289, A303831.

a(n)={binomial(n,2)*2^binomial(n,2)} \\ Andrew Howroyd, Nov 23 2020

a(n) = A000217(n-1) * A006125(n).
a(n) = 2*A095351(n). - Andrew Howroyd, Nov 23 2020
a(n) = A036289(n*(n-1)/2). - Michael Somos, Feb 28 2021

Name replaced by a formula, a(1) changed from 1 to 0, and entry edited by Andrey Zabolotskiy, Jun 05 2022

A277219 Triangle read by rows: T(n,k) is the number of independent sets of size k over all simple labeled graphs on n nodes, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 8, 24, 12, 1, 64, 256, 192, 32, 1, 1024, 5120, 5120, 1280, 80, 1, 32768, 196608, 245760, 81920, 7680, 192, 1, 2097152, 14680064, 22020096, 9175040, 1146880, 43008, 448, 1, 268435456, 2147483648, 3758096384, 1879048192, 293601280, 14680064, 229376, 1024, 1

Offset: 0

Geoffrey Critzer, Oct 05 2016

Equivalently, T(n,k) is the number of size k cliques over all simple labeled graphs on n vertices.

			Triangle begins:
1;
1,     1;
2,     4,      1;
8,     24,     12,     1;
64,    256,    192,    32,    1;
1024,  5120,   5120,   1280,  80,   1;
32768, 196608, 245760, 81920, 7680, 192, 1;
...

Robert Israel, Table of n, a(n) for n = 0..3402 (rows 0 to 81, flattened)

Cf. A079491 (row sums), A006125 (column k=0), A095340 (column k=1), A095351 (column k = 2).

seq(seq(2^(n*(n-1)/2-k*(k-1)/2)*binomial(n,k),k=0..n),n=0..10); # Robert Israel, Oct 06 2016

Table[Table[2^Binomial[n, 2] Binomial[n, k]/2^Binomial[k, 2], {k, 0, n}], {n,0, 7}] // Grid

T(n,k) = 2^binomial(n,2)*binomial(n,k)/2^binomial(k,2).

A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 2, 6, 6, 12, 32, 24, 48, 152, 320, 160, 240, 760, 3640, 6144, 1920, 1920, 4560, 21840, 160224, 229376, 43008, 26880, 42560, 152880, 1121568, 13063792, 16777216, 1835008, 688128, 680960, 1630720, 8972544, 104510336, 2012388736

Offset: 1

Geoffrey Critzer, Aug 26 2013

Row sums = A095340.
Column 1 = A123903.
T(n,k)   = A223894(n,k)*k.
Diagonal = A053549.

			1;
2,    2;
6,    6,    12;
32,   24,   48,    152;
320,  160,  240,   760,    3640;

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.

Alois P. Heinz, Rows n = 1..45, flattened
Index entries for sequences related to graphs

Cf. A070166.

b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
      add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
    end:
T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 26 2013

nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
  1]; Table[
  Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
   1, 7}] // Grid

T(n,k) = binomial(n,k)*k*A001187(k)*A006125(n-k).

A245235 Repeat 2^(n*(n+1)/2) n+1 times.

Original entry on oeis.org

1, 2, 2, 8, 8, 8, 64, 64, 64, 64, 1024, 1024, 1024, 1024, 1024, 32768, 32768, 32768, 32768, 32768, 32768, 2097152, 2097152, 2097152, 2097152, 2097152, 2097152, 2097152, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456, 268435456

Offset: 0

Paul Curtz, Jul 14 2014

For a(n), the successive exponents of 2 are 0, 1, 1, 3, 3, 3,... = A057944(n).

			n+1 times repeated 2^(n*(n+1)/2)= 1, 2, 8, 64, 1024,... = A139685(n).
By the formula: a(0)=1/1=1, a(1)=2/1=2, a(2)=4/2=2, a(3)=8/1=8, a(4)=16/2=8,...
As triangle:
   1,
   2,    2,
   8,    8,    8,
  64,   64,   64,   64,
1024, 1024, 1024, 1024, 1024,
etc.
Row sums: 1, 4, 24, 256,... = A095340.

Cf. A000079, A006125, A057944, A059268, A095340, A123903, A139685.

Table[2^(n*(n+1)/2), {n, 0, 7}, {n+1}] // Flatten (* Jean-François Alcover, Jul 15 2014 *)

from math import isqrt
def A245235(n): return 1<<((m:=isqrt(n+1<<3)-1>>1)*(m+1)>>1) # Chai Wah Wu, Dec 17 2024

a(n) = 2^n/A059268(n).

T(n, k) = 2^(n*(n+1)/2), 0 <= k <= n. - Michel Marcus, Jul 17 2014

A360603 Triangle read by rows. T(n, k) = A360604(n, k) * A001187(k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 4, 0, 8, 6, 12, 38, 0, 64, 32, 48, 152, 728, 0, 1024, 320, 320, 760, 3640, 26704, 0, 32768, 6144, 3840, 6080, 21840, 160224, 1866256, 0, 2097152, 229376, 86016, 85120, 203840, 1121568, 13063792, 251548592

Offset: 0

Peter Luschny, Feb 20 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 0,       1;
[2] 0,       1,      1;
[3] 0,       2,      2,     4;
[4] 0,       8,      6,    12,    38;
[5] 0,      64,     32,    48,   152,    728;
[6] 0,    1024,    320,   320,   760,   3640,   26704;
[7] 0,   32768,   6144,  3840,  6080,  21840,  160224,  1866256;
[8] 0, 2097152, 229376, 86016, 85120, 203840, 1121568, 13063792, 251548592.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.

Algorithms for Competitive Programming, Counting labeled graphs, cp-algorithms.
Albert Nijenhuis and Herbert S Wilf, The enumeration of connected graphs and linked diagrams, Journal of Combinatorial Theory, Vol. 27(3), 1979, Pages 356-359.
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Labeled Graph.

Cf. A006125 Graphs on n labeled nodes, T(n+1, 1) and Sum_{k=0..n} T(n, k).
Cf. A054592 Disconnected labeled graphs with n nodes, Sum_{k=0..n-1} T(n, k).
Cf. A001187 Connected labeled graphs with n nodes, T(n, n).
Cf. A123903 Isolated nodes in all simple labeled graphs on n nodes, T(n+2, 2).
Cf. A053549 Labeled rooted connected graphs, T(n+1, n).
Cf. A275462 Leaves in all simple labeled connected graphs on n nodes T(n+2, n).
Cf. A060818 gcd_{k=0..n} T(n, k) = gcd(n!, 2^n).
Cf. A143543 Labeled graphs on n nodes with k connected components.
Cf. A095340 Total number of nodes in all labeled graphs on n nodes.
Cf. A360604, A360860 (accumulation triangle).

T := (n, k) -> 2^binomial(n-k, 2)*binomial(n-1, k-1)*A001187(k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Based on the recursion:
Trow := proc(n) option remember; if n = 0 then return [1] fi;
seq(2^binomial(n-k, 2) * binomial(n-1, k-1) * Trow(k)[k+1], k = 1..n-1);
2^(n*(n-1)/2) - add(j, j = [%]); [0, %%, %] end:
seq(print(Trow(n)), n = 0..8);

A001187[n_] := A001187[n] = 2^((n - 1)*n/2) - Sum[Binomial[n - 1, k]*2^((k - n + 1)*(k - n + 2)/2)*A001187[k + 1], {k, 0, n - 2}];
T[n_, k_] := 2^Binomial[n - k, 2]*Binomial[n - 1, k - 1]*A001187[k];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 02 2023, after Peter Luschny in A001187 *)

from math import comb as binomial
from functools import cache
@cache
def A360603Row(n: int) -> list[int]:
    if n == 0: return [1]
    s = [2 ** (((k - n + 1) * (k - n)) // 2) * binomial(n - 1, k - 1) * A360603Row(k)[k] for k in range(1, n)]
    b = 2 ** (((n - 1) * n) // 2) - sum(s)
    return [0] + s + [b]

T(n, k) = 2^binomial(n-k, 2)*binomial(n-1, k-1) * A001187(k).
Recursion over the rows of the triangle: Set row(0) = [1] where [.] denotes a 0-based list. Assume now all rows(j) for j < n computed, next compute r = [2^binomial(n-k, 2) * binomial(n-1, k-1) * row(k)[k] for k = 1..n-1] and s = 2^(n*(n-1)/2) - Sum(r). Then row(n) = [0] & r & [s], where '&' denotes the concatenation of lists. (See the Python program for an implementation.)
T(n, n) = A001187(n) (connected labeled graphs).
T(n-1, n) = A053549(n-1) for n >= 1 (labeled rooted connected graphs).
T(n, 1) = Sum_{k>=0} T(n-1, k) = A006125(n-1) for n >= 1 (all labeled graphs).
Sum_{k=0..n-1} T(n, k) = A054592(n) for n >= 1 (disconnected labeled graphs).
See additional formulas in the cross-references.

A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 8, 24, 24, 6, 64, 256, 384, 192, 24, 1024, 5120, 10240, 7680, 1920, 120, 32768, 196608, 491520, 491520, 184320, 23040, 720, 2097152, 14680064, 44040192, 55050240, 27525120, 5160960, 322560, 5040, 268435456, 2147483648, 7516192768, 11274289152, 7046430720, 1761607680, 165150720, 5160960, 40320

Offset: 0

Thomas Scheuerle, Sep 14 2023

A tournament is a directed digraph obtained by assigning a direction for each edge in an undirected complete graph. In a transitive tournament all nodes can be strictly ordered by their reachability.

			Triangle begins:
     1
     1,     1
     2,     4,     2
     8,    24,    24,     6
    64,   256,   384,   192,    24
  1024,  5120, 10240,  7680,  1920,  120

Paul Erdős, On a problem in graph theory, The Mathematical Gazette, 47: 220-223 (1963).

Cf. A002866, A006125, A052670, A095340, A103904, A008279, A117260, A379614 (row sums).

Cf. A122027, A224886, A259105, A350608, A350609, A350610.

T := (n, k) -> 2^(((n-1)*n - (k-1)*k)/2) * n! / (n-k)!:
seq(seq(T(n, k), k = 0..n), n = 0..8);  # Peter Luschny, Nov 02 2023

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2))

T(n, k) = binomial(n, k)*k!*2^(binomial(n, 2) - binomial(k, 2)).
T(n, 0) = A006125(n).
T(n, 1) = A095340(n).
T(n, 2) = A103904(n).
T(n, n) = n!.
T(n, n-1) = A002866(n-1).
T(n, n-2) = A052670(n).
T(n, k) = A008279(n, k) * A117260(n, k). - Peter Luschny, Dec 31 2024

A128406 a(n) = (n+1)2^(n(n+1)).

Original entry on oeis.org

1, 8, 192, 16384, 5242880, 6442450944, 30786325577728, 576460752303423488, 42501298345826806923264, 12379400392853802748991242240, 14278816360970775978458864905355264, 65334214448820184984967924626899496599552, 1187470080331358621040493926581979953470445191168

Offset: 0

Paul Barry, Mar 01 2007

Hankel transform of A069723.

With offset 1, a(n) is the number of vertices with in-degree = 0 over all labeled digraphs (with self loops allowed) on n vertices. Equivalently, the number of elements in all labeled relations on an n-set that have no preimage. - Geoffrey Critzer, Aug 16 2016

Cf. A006125, A069723, A095340.

Table[n 2^(n - 1) 2^(n - 1)^2, {n, 1, 10}]
Table[(n+1)2^(n(n+1)),{n,0,20}] (* Harvey P. Dale, Jun 21 2021 *)

a(n) = A095340(n)*A006125(n+1).

A285529 Triangle read by rows: T(n,k) is the number of nodes of degree k counted over all simple labeled graphs on n nodes, n>=1, 0<=k<=n-1.

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 32, 96, 96, 32, 320, 1280, 1920, 1280, 320, 6144, 30720, 61440, 61440, 30720, 6144, 229376, 1376256, 3440640, 4587520, 3440640, 1376256, 229376, 16777216, 117440512, 352321536, 587202560, 587202560, 352321536, 117440512, 16777216

Offset: 1

Geoffrey Critzer, Apr 20 2017

			1,
2,   2,
6,   12,   6,
32,  96,   96,   32,
320, 1280, 1920, 1280, 320,
...

Row sums give A095340.

Columns for k=0-3: A123903, A095338, A038094, A038096.

nn = 9; Map[Select[#, # > 0 &] &,
  Drop[Transpose[Table[A[z_] := Sum[Binomial[n, k] 2^Binomial[n, 2] z^n/n!, {n, 0, nn}];Range[0, nn]! CoefficientList[Series[z A[z], {z, 0, nn}], z], {k,0, nn - 1}]], 1]] // Grid

E.g.f. for column k: x * Sum_{n>=0} binomial(n,k)*2^binomial(n,2)*x^n/n!.
Sum_{k=1..n-1} T(n,k)*k/2 = A095351(n).
T(n,k) = n*binomial(n-1,k)*2^binomial(n-1,2). - Alois P. Heinz, Apr 21 2017

cp's OEIS Frontend

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

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A103904 a(n) = n*(n-1)/2 * 2^(n*(n-1)/2).

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A277219 Triangle read by rows: T(n,k) is the number of independent sets of size k over all simple labeled graphs on n nodes, n>=0, 0<=k<=n.

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A228315 Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.

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A245235 Repeat 2^(n*(n+1)/2) n+1 times.

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A360603 Triangle read by rows. T(n, k) = A360604(n, k) * A001187(k) for 0 <= k <= n.

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A365638 Triangular array read by rows: T(n, k) is the number of ways that a k-element transitive tournament can occur as a subtournament of a larger tournament on n labeled vertices.

A103904 a(n) = n(n-1)/2 2^(n*(n-1)/2).

A128406 a(n) = (n+1)2^(n(n+1)).