A006125 -id:A006125 - cp's OEIS frontend

A004003 Number of domino tilings (or dimer coverings) of a 2n X 2n square.

1, 2, 36, 6728, 12988816, 258584046368, 53060477521960000, 112202208776036178000000, 2444888770250892795802079170816, 548943583215388338077567813208427340288, 1269984011256235834242602753102293934298576249856

Offset: 0

N. J. A. Sloane, Greg Huber

A099390 is the main entry for domino tilings (or dimer tilings) of a rectangle.
The numbers of domino tilings in A006253, A004003, A006125 give 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
Christine Bessenrodt points out that Pachter (1997) shows that a(n) is divisible by 2^n (cf. A065072).
a(n) is the number of different ways to cover a 2n X 2n lattice with 2n^2 dominoes. John and Sachs show that a(n) = 2^n*B(n)^2, where B(n) == n+1 (mod 32) when n is even and B(n) == (-1)^((n-1)/2)*n (mod 32) when n is odd. - Yong Kong (ykong(AT)curagen.com), May 07 2000

			The 36 solutions for the 4 X 4 board, from Neven Juric, May 14 2008:
A01 = {(1,2), (3,4), (5,6), (7,8), (9,10), (11,12), (13,14), (15,16)}
A02 = {(1,2), (3,4), (5,6), (7,11), (9,10), (8,12), (13,14), (15,16)}
A03 = {(1,2), (3,4), (5,9), (6,7), (10,11), (8,12), (13,14), (15,16)}
A04 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,12), (13,14), (15,16)}
A05 = {(1,2), (3,4), (5,9), (6,10), (7,11), (8,12), (13,14), (15,16)}
A06 = {(1,2), (3,4), (5,6), (7,8), (9,10), (13,14), (11,15), (12,16)}
A07 = {(1,2), (3,4), (5,9), (6,10), (7,8), (11,15), (13,14), (12,16)}
A08 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,12), (15,16)}
A09 = {(1,2), (3,4), (5,6), (7,11), (8,12), (9,13), (10,14), (15,16)}
A10 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,11), (14,15), (12,16)}
A11 = {(1,2), (3,4), (5,6), (7,8), (9,13), (10,14), (11,15), (12,16)}
A12 = {(1,2), (5,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A13 = {(1,2), (3,7), (4,8), (5,9), (6,10), (11,12), (13,14), (15,16)}
A14 = {(1,2), (5,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}
A15 = {(1,2), (3,7), (4,8), (6,10), (5,9), (11,15), (12,16), (13,14)}
A16 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,14), (11,12), (15,16)}
A17 = {(1,2), (3,7), (4,8), (5,6), (9,13), (10,11), (14,15), (12,16)}
A18 = {(1,2), (5,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A19 = {(1,5), (2,6), (3,4), (7,8), (9,10), (11,12), (13,14), (15,16)}
A20 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,10), (13,14), (15,16)}
A21 = {(1,5), (3,4), (2,6), (9,10), (7,8), (11,15), (13,14), (12,16)}
A22 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,12), (15,16)}
A23 = {(1,5), (2,6), (3,4), (7,11), (8,12), (9,13), (10,14), (15,16)}
A24 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,11), (14,15), (12,16)}
A25 = {(1,5), (2,6), (3,4), (7,8), (9,13), (10,14), (11,15), (12,16)}
A26 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A27 = {(1,5), (2,6), (3,7), (4,8), (9,10), (11,12), (13,14), (15,16)}
A28 = {(1,5), (2,3), (6,7), (4,8), (9,10), (11,15), (13,14), (12,16)}
A29 = {(1,5), (2,6), (3,7), (4,8), (9,10), (13,14), (11,15), (12,16)}
A30 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,12), (15,16)}
A31 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,12), (15,16)}
A32 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A33 = {(1,5), (2,6), (3,7), (4,8), (9,13), (10,11), (14,15), (12,16)}
A34 = {(1,5), (2,3), (4,8), (6,10), (7,11), (9,13), (14,15), (12,16)}
A35 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,14), (11,15), (12,16)}
A36 = {(1,5), (2,3), (6,7), (4,8), (9,13), (10,11), (14,15), (12,16)}

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 569.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Darko Veljan, Kombinatorika s teorijom grafova (Croatian) (Combinatorics with Graph Theory) mentions the value 12988816 = 2^4*901^2 for the 8 X 8 case on page 4.

Alois P. Heinz, Table of n, a(n) for n = 0..44 (first 26 terms from T. D. Noe)
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
N. Allegra, Exact solution of the 2d dimer model: Corner free energy, correlation functions and combinatorics, arXiv:1410.4131 [cond-mat.stat-mech], 2014.
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, Journal of Combinatorial Theory, Series A, Volume 77, Issue 1, January 1997, Pages 67-97.
H. Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
Steven R. Finch, The Dimer Problem [From Steven Finch, Apr 20 2019]
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 363
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Illustration for a(2) = 36 [Fig. 9 from "Sandpiles and Dominos"]
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, Discrete Math. 216 (2000), no. 1-3, 211-219.
Adrien Kassel, Le modèle de dimères, Images des Mathématiques, CNRS, 2016. [In French]
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Viet-Ha Nguyen, Kévin Perrot, and Mathieu Vallet, NP-completeness of the game Kingdomino(TM), Theoretical Computer Science (2020) Vol. 822, 23-35. See also arXiv:1909.02849, [cs.CC], 2019.
Lior Pachter, Combinatorial approaches and conjectures ..., Elec. J. Comb. 4 (1997) #R29.
James Propp, Some 2-Adic Conjectures Concerning Polyomino Tilings of Aztec Diamonds, Integers (2023) Vol. 23, Art. A30.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
N. J. A. Sloane, Illustration for a(2) = 36 [Slide from an old talk I gave]
R. P. Stanley, A combinatorial miscellany
Eric Weisstein's World of Mathematics, Domino Tiling
Eric Weisstein, Illustration for a(2) = 36, from Domino Tilings web page (see previous link) [Included with permission]
Index entries for sequences related to dominoes

Cf. A028420, A006253, A006125, A065072, A124997, A239273, A256043.

Main diagonal of array A099390 or A187596.

f := n->2^(2*n^2)*product(product(cos(i*Pi/(2*n+1))^2+cos(j*Pi/(2*n+1))^2,j=1..n),i=1..n); for k from 0 to 12 do round(evalf(f(k),300)) od;

a[n_] := Round[ N[ Product[ 2*Cos[(2i*Pi)/(2n+1)] + 2*Cos[(2j*Pi)/(2n+1)] + 4,  {i, 1, n}, {j, 1, n}], 300] ]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 04 2012, after Maple *)
Table[Sqrt[Resultant[ChebyshevU[2*n, x/2], ChebyshevU[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Dec 30 2020 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(2*n, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

from math import isqrt
from sympy.abc import x
from sympy import I, resultant, chebyshevu
def A004003(n): return isqrt(resultant(chebyshevu(n<<1,x/2),chebyshevu(n<<1,I*x/2))) if n else 1 # Chai Wah Wu, Nov 07 2023

a(n) = A099390(2n,2n).
a(n) = Product_{j=1..n} Product_{k=1..n} (4*cos(j*Pi/(2*n+1))^2 + 4*cos(k*Pi/(2*n+1))^2). - N. J. A. Sloane, Mar 16 2015
a(n) = 2^n * A065072(n)^2. - Alois P. Heinz, Nov 22 2018
a(n)^2 = Resultant(U(2*n,x/2), U(2*n,i*x/2)), where U(n,x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Apr 13 2020
a(n) ~ 2 * (sqrt(2)-1)^(2*n+1) * exp(G*(2*n+1)^2/Pi), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020

Corrected and extended by David Radcliffe

A016031 De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct.

Original entry on oeis.org

1, 1, 2, 16, 2048, 67108864, 144115188075855872, 1329227995784915872903807060280344576, 226156424291633194186662080095093570025917938800079226639565593765455331328

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Sequence corresponds also to the largest number that may be determined by asking no more than 2^(n-1) - 1 questions("Yes"-or-"No" answerable) with lying allowed at most once. - Lekraj Beedassy, Jul 15 2002
Number of Ouroborean rings for binary n-tuplets. - Lekraj Beedassy, May 06 2004
Also the number of games of Nim that are wins for the second player when the starting position is either the empty heap or heaps of sizes 1 <= t_1 < t_2 < ... < t_k < 2^(n-1) (if n is 1, the only starting position is the empty heap). E.g.: a(4) = 16: the winning positions for the second player when all the heap sizes are different and less than 2^3: (4,5,6,7), (3,5,6), (3,4,7), (2,5,7), (2,4,6), (2,3,6,7), (2,3,4,5), (1,6,7), (1,4,5), (1,3,5,7), (1,3,4,6), (1,2,5,6), (1,2,4,7), (1,2,3), (1,2,3,4,5,6,7) and the empty heap. - Kennan Shelton (kennan.shelton(AT)gmail.com), Apr 14 2006
a(n + 1) = 2^(2^n-n-1) = 2^A000295(n) is also the number of set-systems on n vertices with no singletons. The case with singletons is A058891. The unlabeled case is A317794. The spanning/covering case is A323816. The antichain case is A006126. The connected case is A323817. The uniform case is A306021(n) - 1. The graphical case is A006125. The chain case is A005840. - Gus Wiseman, Feb 01 2019
Named after the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - Amiram Eldar, Jun 20 2021

Jonathan L. Gross and Jay Yellen, eds., Handbook of Graph Theory, CRC Press, 2004, p. 255.
Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973, p. 31.
D. J. Newman, "A variation of the Game of Twenty Question", in: Murray S. Klamkin (ed.), Problems in Applied Mathematics, Philadelphia, PA: SIAM, 1990, Problem 66-20, pp. 121-122.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.6.15.
Ian Stewart, Game, Set and Math, pp. 50, Penguin 1991.

Vincenzo Librandi, Table of n, a(n) for n = 1..12
CombOS - Combinatorial Object Server, Generate de Bruijn sequences.
Robert Erra, Nik Lygeros and Nigel Stewart, On Minimal Strings Containing the Elements of S(n) by Decimation, Proceedings AA (DM-CCG), 2001, Section 5.4.
Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.
Francisco J. Muñoz and Juan Carlos Nuño, Rule-based Generation of de Bruijn Sequences: Memory and Learning, arXiv:2507.09764 [cs.FL], 2025. See p. 9.
Wikipedia, De Bruijn sequence.

Cf. A000295, A003465, A006125, A058891 (set systems), A317794 (unlabeled case), A323816 (spanning case), A323817 (connected case), A331691 (alternating signs).

[2^(2^(n-1)-n): n in [1..10]]; // Vincenzo Librandi, Aug 09 2017

Table[2^(2^(n - 1) - n), {n, 20}] (* Vincenzo Librandi, Aug 09 2017 *)

a(n) = 2^A000295(n-1). - Lekraj Beedassy, Jan 17 2007

Shifting once to the left gives the binomial transform of A323816. - Gus Wiseman, Feb 01 2019

A116508 a(n) = C( C(n,2), n).

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 5005, 116280, 3108105, 94143280, 3190187286, 119653565850, 4922879481520, 220495674290430, 10682005290753420, 556608279578340080, 31044058215401404845, 1845382436487682488000, 116475817125419611477660, 7779819801401934344268210

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Mar 21 2006

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - Gus Wiseman, Dec 22 2023

			a(5) = C(C(5,2),5) = C(10,5) = 252.

Alois P. Heinz, Table of n, a(n) for n = 0..370

Main diagonal of A084546.
The unlabeled version is A001434, covering case A006649.
The connected case is A057500, unlabeled A001429.
For set-systems we have A136556, covering case A054780.
The covering case is A367863.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A133686 counts graphs satisfying a strict AOC, connected A129271.
A367867 counts graphs contradicting a strict AOC, connected A140638.
Cf. A001187, A003465, A143543, A305000, A367916, A367917.

[0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // Vincenzo Librandi, Nov 03 2014

a:= n-> binomial(binomial(n, 2), n):
seq(a(n), n=0..20);

nn = 18; f[x_, y_] :=
Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[
n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* Geoffrey Critzer, Nov 02 2014 *)
Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)
Table[SeriesCoefficient[(1 + x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

from math import comb
def A116508(n): return comb(n*(n-1)>>1,n) # Chai Wah Wu, Jul 02 2024

[(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # Zerinvary Lajos, Nov 30 2009

a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2020

a(n) = [x^n] (1+x)^(n*(n-1)/2). - Vaclav Kotesovec, Aug 06 2025

a(0)=1 prepended by Alois P. Heinz, Feb 02 2024

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

A062740 Number of connected labeled graphs with loops.

Original entry on oeis.org

1, 2, 4, 32, 608, 23296, 1709056, 238880768, 64396439552, 33943701028864, 35324404321091584, 72994114660256448512, 300460426062916084563968, 2468021884106048216693211136, 40495494119922790159005962469376, 1328011048967552376327692463141552128

Offset: 0

Vladeta Jovovic, Jul 12 2001

Alois P. Heinz, Table of n, a(n) for n = 0..80

Cf. A006125, A226773.

logtr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else p(n)- add(k *binomial(n, k) *p(n-k) *b(k), k=1..n-1)/n fi end end:
a:= logtr(n-> 2^binomial(n+1, 2)):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 01 2014

nn=14;g=Sum[2^Binomial[n,2](2x)^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[g]+1,{x,0,nn}],x] (* Geoffrey Critzer, Feb 01 2014 *)

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

E.g.f.: A(2*x) where A(x) is the e.g.f. for A001187. - Geoffrey Critzer, Feb 01 2014

A003216 Number of Hamiltonian graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 8, 48, 383, 6196, 177083, 9305118, 883156024, 152522187830, 48322518340547

Offset: 1

N. J. A. Sloane

a(1) could also be taken to be 0, but I prefer a(1) = 1. - N. J. A. Sloane, Oct 15 2006

J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Asplund, N. Bradley Fox, and Arran Hamm, New Perspectives on Neighborhood-Prime Labelings of Graphs, arXiv:1804.02473 [math.CO], 2018.
J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271. (Annotated scanned copy of 3 pages)
Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, preprint, 2015.
Scott Garrabrant and Igor Pak, Pattern Avoidance is Not P-Recursive, arXiv:1505.06508 [math.CO], 2015.
Jan Goedgebeur, Barbara Meersman, and Carol T. Zamfirescu, Graphs with few Hamiltonian Cycles, arXiv:1812.05650 [math.CO], 2018-2019.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Hamiltonian Graph
Wikipedia, Hamiltonian path
Gus Wiseman, Non-isomorphic representatives of the a(5) = 8 simple graphs containing a Hamiltonian cycle.

Main diagonal of A325455 and of A325447 (for n>=3).
The labeled case is A326208.
The directed case is A326226 (with loops) or A326225 (without loops).
The case without loops is A326215.
Unlabeled simple graphs not containing a Hamiltonian cycle are A246446.
Unlabeled simple graphs containing a Hamiltonian path are A057864.
Cf. A000088, A006125, A283420.

A000088(n) = a(n) + A246446(n). - Gus Wiseman, Jun 17 2019

Extended to n=11 by Brendan McKay, Jul 15 1996
a(12) from Sean A. Irvine, Mar 17 2015
a(13) from A246446 added by Jan Goedgebeur, Sep 07 2019

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

Original entry on oeis.org

0, 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 497665, 2128127, 9183489, 39940863, 174897665, 770452479, 3411959809, 15181264895, 67833868289, 304256253951, 1369404661761, 6182858317823, 27995941060609, 127100310290431, 578433619525633, 2638370120138751

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple context-free grammar.
Number of lattice paths from (0,0) to (2n-2,0) that stay (weakly) in the first quadrant and such that each step is either U=(1,1), D=(1,-1), or L=(3,1). Equivalently, underdiagonal lattice paths from (0,0) to (n-1,n-1) and such that each step is either (1,0), (0,1), or (2,1). E.g., a(4)=9 because in addition to the five Dyck paths from (0,0) to (6,0) [UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD] we have LDUD, LUDD, ULDD and UDLD. - Emeric Deutsch, Dec 21 2003
Hankel transform of a(n+1) is A006125(n+1). - Paul Barry, Apr 01 2007
Also, a(n+1) is the number of walks from (0,0) to (n,0) using steps (1,1), (1,-1) and (0,-1). See the U(n,k) array in A071943, where A052709(n+1) = U(n,0). - N. J. A. Sloane, Mar 29 2013
Diagonal sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
From Gus Wiseman, Jun 17 2021: (Start)
Conjecture: For n > 0, also the number of sequences of length n - 1 covering an initial interval of positive integers and avoiding three terms (..., x, ..., y, ..., z, ...) such that x <= y <= z. The version avoiding the strict pattern (1,2,3) is A226316. Sequences covering an initial interval are counted by A000670. The a(1) = 1 through a(4) = 9 sequences are:
  ()  (1)  (1,1)  (1,2,1)
           (1,2)  (1,3,2)
           (2,1)  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..499
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, Jumping succession rules and their generating functions, Discrete Math., 271 (2003), 29-50.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 664
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - _N. J. A. Sloane_, Dec 27 2012
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Diagonal entries of A071943 and A071945.
Cf. A000108, A000670, A025227, A052709, A056986, A071943, A085880.
Cf. A102726, A117434, A158005, A226316, A333217, A335479.

[0] cat [(&+[Binomial(n,k+1)*Binomial(2*k,n-1): k in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, May 30 2022

spec := [S,{C=Prod(B,Z),S=Union(B,C,Z),B=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[(y-y^2)/(1+y^2), {y, 0, 24}], x] (* then A(x)= y(x) *) (* Len Smiley, Apr 12 2000 *)
CoefficientList[Series[(1 -Sqrt[1 -4x -4x^2])/(2(1+x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 12 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1+x+O(x^n))))/2/(1+x),n)

[sum(binomial(k, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 30 2022

a(n) + a(n-1) = A025227(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-2-2*k)!/(k!*(n-k)!*(n-1-2*k)!). - Emeric Deutsch, Nov 14 2001
D-finite with recurrence: n*a(n) = (3*n-6)*a(n-1) + (8*n-18)*a(n-2) + (4*n-12)*a(n-3), n>2. a(1)=a(2)=1.
a(n) = b(1)*a(n-1) + b(2)*a(n-2) + ... + b(n-1)*a(1) for n>1 where b(n)=A025227(n).
G.f.: A(x) = x/(1-(1+x)*A(x)). - Paul D. Hanna, Aug 16 2002
G.f.: A(x) = x/(1-z/(1-z/(1-z/(...)))) where z=x+x^2 (continued fraction). - Paul D. Hanna, Aug 16 2002; revised by Joerg Arndt, Mar 18 2011
a(n+1) = Sum_{k=0..n} Catalan(k)*binomial(k, n-k). - Paul Barry, Feb 22 2005
From Paul Barry, Mar 14 2006: (Start)
G.f. is x*c(x*(1+x)) where c(x) is the g.f. of A000108.
Row sums of A117434. (End)
a(n+1) = (1/(2*Pi))*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*(4+4x-x^2)/(2*(1+x)). - Paul Barry, Apr 01 2007
From Gary W. Adamson, Jul 22 2011: (Start)
For n>0, a(n) is the upper left term in M^(n-1), where M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  ... (End)
G.f.: x*Q(0), where Q(k) = 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ sqrt(2-sqrt(2))*2^(n-1/2)*(1+sqrt(2))^(n-1)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
a(n+1) = Sum_{k=0..floor(n/2)} A085880(n-k,k). - Philippe Deléham, Nov 15 2013

Better g.f. and recurrence from Michael Somos, Aug 03 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A098844 a(1)=1, a(n) = n*a(floor(n/2)).

Original entry on oeis.org

1, 2, 3, 8, 10, 18, 21, 64, 72, 100, 110, 216, 234, 294, 315, 1024, 1088, 1296, 1368, 2000, 2100, 2420, 2530, 5184, 5400, 6084, 6318, 8232, 8526, 9450, 9765, 32768, 33792, 36992, 38080, 46656, 47952, 51984, 53352, 80000, 82000, 88200, 90300

Offset: 1

Benoit Cloitre, Nov 03 2004

nonn

			a(10) = floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3) = 10*5*2*1 = 100;
a(17) = 1088 since 17 = 10001(base 2) and so a(17) = 10001*1000*100*10*1(base 2) = 17*8*4*2*1 = 1088.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.
For formulas regarding a general parameter p (i.e., terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=3 to p=12 see A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

lst={};Do[p=n;s=1;While[p>1,p=IntegerPart[p/2];s*=p;];AppendTo[lst,s],{n,1,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI
```
a(n)=if(n<2,1,n*a(floor(n/2)))
```

from math import prod
def A098844(n): return n*prod(n//2**k for k in range(1,n.bit_length()-1)) # Chai Wah Wu, Jun 07 2022

a(2^n) = 2^(n*(n+1)/2) = A006125(n+1).
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = Product_{k=0..floor(log_2(n))} floor(n/2^k), n>=1.
Recurrence:
a(n*2^m) = n^m*2^(m(m+1)/2)*a(n).
a(n) <= n^((1+log_2(n))/2) = 2^A000217(log_2(n)); equality iff n is a power of 2.
a(n) >= c(n)*(n+1)^((1 + log_2(n+1))/2) for n != 2,
where c(n) = Product_{k=1..floor(log_2(n))} (1 - 1/2^k); equality holds iff n+1 is a power of 2.
a(n) > c*(n+1)^((1 + log_2(n+1))/2)
where c = 0.288788095086602421... (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo.
lim sup a(n)/n^((1+log_2(n))/2) = 1 for n-->oo.
lim inf a(n)/a(n+1) = 0.288788095086602421... for n-->oo (see constant A048651).
a(n) = O(n^((1+log_2(n))/2)). (End)

Formula section edited by Hieronymus Fischer, Jun 13 2012

A129271 Number of labeled n-node connected graphs with at most one cycle.

Original entry on oeis.org

1, 1, 1, 4, 31, 347, 4956, 85102, 1698712, 38562309, 980107840, 27559801736, 849285938304, 28459975589311, 1030366840792576, 40079074477640850, 1666985134587145216, 73827334760713500233, 3468746291121007607808, 172335499299097826575564, 9027150377126199463936000

Offset: 0

Washington Bomfim, May 10 2008

The majority of those graphs of order 4 are trees since we have 16 trees and only 9 unicycles. See example.

Also connected graphs covering n vertices with at most n edges. The unlabeled version is A005703. - Gus Wiseman, Feb 19 2024

			a(4) = 16 + 3*3 = 31.
From _Gus Wiseman_, Feb 19 2024: (Start)
The a(0) = 1 through a(3) = 4 graph edge sets:
  {}  .  {{1,2}}  {{1,2},{1,3}}
                  {{1,2},{2,3}}
                  {{1,3},{2,3}}
                  {{1,2},{1,3},{2,3}}
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Dover, 2002, p. 2.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Wikipedia, PseudoForest.

Cf. A129137, A000272.
For any number of edges we have A001187, unlabeled A001349.
The unlabeled version is A005703.
The case of equality is A057500, covering A370317, cf. A370318.
The non-connected non-covering version is A133686.
The connected complement is A140638, unlabeled A140636, covering A367868.
The non-connected covering version is A367869 or A369191.
The version with loops is A369197, non-connected A369194.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by number of edges.
Cf. A006649, A116508, A134964, A143543, A323818, A367862, A367863, A367867, A367916, A367917, A368951.

a := n -> `if`(n=0,1,((n-1)*exp(n)*GAMMA(n-1,n)+n^(n-2)*(3-n))/2):
seq(simplify(a(n)),n=0..16); # Peter Luschny, Jan 18 2016

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[ Log[1/(1-t)]/2+t/2-3t^2/4+1,{x,0,nn}],x]  (* Geoffrey Critzer, Mar 23 2013 *)

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + t/2 - 3*t^2/4 + 1))} \\ Andrew Howroyd, Nov 07 2019

a(0) = 1, for n >=1, a(n) = A000272(n) + A057500(n) = n^{n-2} + (n-1)(n-2)/2Sum_{r=1..n-2}( (n-3)!/(n-2-r)! )n^(n-2-r)
E.g.f.: log(1/(1-T(x)))/2 + T(x)/2 - 3*T(x)^2/4 + 1, where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, Mar 23 2013
a(n) = ((n-1)*e^n*GAMMA(n-1,n)+n^(n-2)*(3-n))/2 for n>=1. - Peter Luschny, Jan 18 2016

Terms a(17) and beyond from Andrew Howroyd, Nov 07 2019

A001858 Number of forests of trees on n labeled nodes.

Original entry on oeis.org

1, 1, 2, 7, 38, 291, 2932, 36961, 561948, 10026505, 205608536, 4767440679, 123373203208, 3525630110107, 110284283006640, 3748357699560961, 137557910094840848, 5421179050350334929, 228359487335194570528, 10239206473040881277575, 486909744862576654283616

Offset: 0

N. J. A. Sloane and Simon Plouffe

The number of integer lattice points in the permutation polytope of {1,2,...,n}. - Max Alekseyev, Jan 26 2010
Equals the number of score sequences for a tournament on n vertices. See Prop. 7 of the article by Bartels et al., or Example 3.1 in the article by Stanley. - David Radcliffe, Aug 02 2022
Number of labeled acyclic graphs on n vertices. The unlabeled version is A005195. The covering case is A105784, connected A000272. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of the a(4) = 38 forests:
  {}  {12}  {12,13}  {12,13,14}
      {13}  {12,14}  {12,13,24}
      {14}  {12,23}  {12,13,34}
      {23}  {12,24}  {12,14,23}
      {24}  {12,34}  {12,14,34}
      {34}  {13,14}  {12,23,24}
            {13,23}  {12,23,34}
            {13,24}  {12,24,34}
            {13,34}  {13,14,23}
            {14,23}  {13,14,24}
            {14,24}  {13,23,24}
            {14,34}  {13,23,34}
            {23,24}  {13,24,34}
            {23,34}  {14,23,24}
            {24,34}  {14,23,34}
                     {14,24,34}
(End)

B. Bollobas, Modern Graph Theory, Springer, 1998, p. 290.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..388 (first 101 terms from T. D. Noe)
J. E. Bartels, J. Mount, and D. J. A. Welsh, The polytope of win vectors. Annals of Combinatorics 1, 1-15 (1997). https://doi.org/10.1007/BF02558460
David Callan, A Combinatorial Derivation of the Number of Labeled Forests, J. Integer Seqs., Vol. 6, 2003.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.o
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002.
Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions, thesis, 2002 [Local copy, with permission]
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 132.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 7, arXiv:1709.08416 [math.CO], 2017.
Anton Izosimov, Matrix polynomials, generalized Jacobians, and graphical zonotopes, arXiv:1506.05179 [math.AG], 2015.
Qipeng Kuang, Ondřej Kuželka, Yuanhong Wang, and Yuyi Wang, Bridging Weighted First Order Model Counting and Graph Polynomials, arXiv:2407.11877 [cs.LO], 2024. See pp. 32-33.
Arun P. Mani and Rebecca J. Stones, Congruences for weighted number of labeled forests, INTEGERS 16 (2016). #A17.
J. Pitman, Coalescent Random Forests, J. Combin. Theory, A85 (1999), 165-193.
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
J. Riordan, Letter to N. J. A. Sloane, Oct 19 1970
J. Riordan, Letter to N. J. A. Sloane, Nov 10 1975
John Riordan and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane, Letter to J. Riordan, Nov. 1970
R. Stanley, Decompositions of rational convex polytopes, Ann. Discrete Math 6.6 (1980): 333-342.
L. Takacs, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581.
E. M. Wright, A relationship between two sequences, Proc. London Math. Soc. (3) 17 (1967) 296-304.
Index entries for sequences related to forests

The connected case is A000272, rooted A000169.
The unlabeled version is A005195, connected A000055.
The covering case is A105784, unlabeled A144958.
Row sums of A138464.
For triangles instead of cycles we have A213434, covering A372168.
For a unique cycle we have A372193, covering A372195.
A002807 counts cycles in a complete graph.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A006572, A006573, A088956.
Cf. A053530, A054548, A137916, A263340, A322661, A367863, A372176.

exp(x+x^2+add(n^(n-2)*x^n/n!, n=3..50));
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*j^(j-2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 15 2008
# third Maple program:
F:= exp(-LambertW(-x)*(1+LambertW(-x)/2)):
S:= series(F,x,51):
seq(coeff(S,x,j)*j!, j=0..50); # Robert Israel, May 21 2015

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[ Series[Exp[t-t^2/2],{x,0,nn}],x] (* Geoffrey Critzer, Sep 05 2012 *)
nmax = 20; CoefficientList[Series[-LambertW[-x]/(x*E^(LambertW[-x]^2/2)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jul 19 2019 *)

a(n)=if(n<0,0,sum(m=0,n,sum(j=0,m,binomial(m,j)*binomial(n-1,n-m-j)*n^(n-m-j)*(m+j)!/(-2)^j)/m!)) /* Michael Somos, Aug 22 2002 */

E.g.f.: exp( Sum_{n>=1} n^(n-2)*x^n/n! ). This implies (by a theorem of Wright) that a(n) ~ exp(1/2)*n^(n-2). - N. J. A. Sloane, May 12 2008 [Corrected by Philippe Flajolet, Aug 17 2008]
E.g.f.: exp(T - T^2/2), where T = T(x) = Sum_{n>=1} n^(n-1)*x^n/n! is Euler's tree function (see A000169). - Len Smiley, Dec 12 2001
Shifts 1 place left under the hyperbinomial transform (cf. A088956). - Paul D. Hanna, Nov 03 2003
a(0) = 1, a(n) = Sum_{j=0..n-1} C(n-1,j) (j+1)^(j-1) a(n-1-j) if n>0. - Alois P. Heinz, Sep 15 2008

More terms from Michael Somos, Aug 22 2002

cp's OEIS Frontend

A004003 Number of domino tilings (or dimer coverings) of a 2n X 2n square.

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A016031 De Bruijn's sequence: 2^(2^(n-1) - n): number of ways of arranging 2^n bits in circle so all 2^n consecutive strings of length n are distinct.

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A116508 a(n) = C( C(n,2), n).

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

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A062740 Number of connected labeled graphs with loops.

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A003216 Number of Hamiltonian graphs with n nodes.

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A052709 Expansion of g.f. (1-sqrt(1-4*x-4*x^2))/(2*(1+x)).

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A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).