A022562 -id:A022562 - cp's OEIS frontend

A005226 Number of atomic species of degree n; also number of connected permutation groups of degree n.

0, 1, 1, 2, 6, 6, 27, 20, 130, 124, 598, 641, 4850, 4772, 35625, 46074, 389839, 487408, 4617554

Offset: 0

An atomic species is one that is not the product of smaller species. - Christian G. Bower, Feb 23 2006

A permutation group is connected if it is not the direct product of smaller permutation groups. - Christian G. Bower, Feb 23 2006

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 147.
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. Decoste, G. Labelle & J. Labelle, Espèces sur les petites cardinalités Tableaux divers, Université du Québec à Montréal (octobre 1988), Unpublished.
Jacques Labelle, Quelques espèces sur les ensembles de petite cardinalité, Ann. Sc. Math. Québec 9.1 (1985): 31-58. (Annotated scanned copy of preprint)
J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.
L. Naughton and G. Pfeiffer, Integer sequences realized by the subgroup pattern of the symmetric group, arXiv:1211.1911 [math.GR], 2012-2013 and J. Int. Seq. 16 (2013) #13.5.8.
N. J. A. Sloane, Transforms

Cf. A005227. Unlabeled version of A116655.

A000638 = Cases[Import["https://oeis.org/A000638/b000638.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{0} ~Join~ EulerInvTransform[A000638 // Rest] (* Jean-François Alcover, Dec 03 2019, updated Mar 17 2020 *)

Inverse Euler transform of A000638. Define b(n), c(n), d(): b(1)=d(1)=0. b(k)=A005227(k), k>1. c(k)=A000638(k), k>0. d(k)=a(k), k>1. d is Dirichlet convolution of b and c. - Christian G. Bower, Feb 23 2006

a(11) corrected and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's edit to A000638.
Could be extended to a(18) now using the new terms for A000637. - N. J. A. Sloane, Jul 30 2010
a(13) from Liam Naughton, Nov 23 2012
a(14)-a(18) from the inverse Euler transform of A000637. - R. J. Mathar, Mar 03 2015

A053465 Number of connected 2-multigraphs on n nodes.

Original entry on oeis.org

1, 1, 2, 7, 53, 712, 24576, 2275616, 589543159, 420188096140, 819411181635025, 4381819315336997184, 64583749250393921183423, 2638507778912832094660037006, 300397569392490080058575760090548, 95776592061550107555640978862165082446

Offset: 0

Vladeta Jovovic, Jan 13 2000

A 2-multigraph is similar to an ordinary graph except there are 0, 1 or 2 edges between any two nodes (self-loops are not allowed).

Also the number of connected signed graphs on n unlabeled nodes. - Andrew Howroyd, Sep 25 2018

Andrew Howroyd, Table of n, a(n) for n = 0..50
Edward A. Bender and E. Rodney Canfield, Enumeration of connected invariant graphs, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 273.

Cf. A004102, A318590.

A004102 = Import["https://oeis.org/A004102/b004102.txt", "Table"][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[A004102 // Rest]] (* Jean-François Alcover, Sep 12 2019, after Andrew Howroyd, updated Mar 17 2020 *)

Inverse Euler transform of A004102. - Andrew Howroyd, Sep 25 2018

a(0)=1 prepended and terms a(15) and beyond from Andrew Howroyd, Sep 25 2018

A022564 Number of 2-connected non-Hamiltonian claw-free graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 16, 84, 408

Offset: 1

N. J. A. Sloane

R. Faudree, E. Flandrin and Z. Ryjacek, Claw-free graphs - a survey, Discr. Math., 164 (1997), 87-147.

Cf. A022562, A022564.

A126757 Number of n-node connected graphs with no cycles of length less than 5.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 18, 47, 137, 464, 1793, 8167, 43645, 275480, 2045279, 17772647, 179593823, 2098423758, 28215583324, 434936005284, 7662164738118

Offset: 1

N. J. A. Sloane, Feb 18 2007

Keith M. Briggs, Combinatorial Graph Theory
CombOS - Combinatorial Object Server, Generate graphs

Cf. A006787, A140440.

A006787 = Cases[Import["https://oeis.org/A006787/b006787.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A006787] (* Jean-François Alcover, May 13 2019, updated Mar 17 2020 *)

This is the inverse Euler transform of A006787. - Conjectured by Vladeta Jovovic, Jun 16 2008, proved by Max Alekseyev and Brendan McKay, Jun 17 2008

Definition corrected by Max Alekseyev and Brendan McKay, Jun 17 2008

a(20)-a(21) using Brendan McKay's extension to A006787 by Alois P. Heinz, Mar 11 2018

A001928 Number of connected topologies with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 6, 21, 94, 512, 3485, 29515, 314474, 4255727, 73831813, 1653083021, 47941962135, 1803010446411, 87882300251730, 5543501326580737

Offset: 0

N. J. A. Sloane

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184.
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).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, personal communication.

K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A001929, A001930.

A001930 = Cases[Import["https://oeis.org/A001930/b001930.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
{1} ~Join~ EulerInvTransform[Rest[A001930]] (* Jean-François Alcover, Jan 01 2020, updated Mar 17 2020 *)

Inverse Euler transform of A001930. - Vladeta Jovovic, Jan 06 2006

More terms from Vladeta Jovovic, Jan 06 2006

A086991 Numbers of claw-free simple graphs (not necessarily connected) on n nodes.

Original entry on oeis.org

1, 2, 4, 10, 26, 85, 302, 1285, 6170, 34294, 227417, 2001617, 26098641, 520051389, 15041205304

Offset: 1

Eric W. Weisstein, Jul 28 2003

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 8489dde.
Avraham Itzhakov and Michael Codish, Breaking Symmetries in Graph Search with Canonizing Sets, arXiv:1511.08205 [cs.AI], 2015
Eric Weisstein's World of Mathematics, Claw-Free Graph

Cf. A022562 (inv. Euler transf.).

a(10) from Michael Codish, Dec 01 2015

a(11)-a(15) added using tinygraph by Falk Hüffner, Jan 12 2016

A349904 Inverse Euler transform of the tribonacci numbers A000073.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 10, 18, 31, 56, 96, 172, 299, 530, 929, 1646, 2893, 5126, 9044, 16028, 28362, 50328, 89249, 158598, 281830, 501538, 892857, 1591282, 2837467, 5064334, 9044023, 16163946, 28906213, 51729844, 92628401, 165967884, 297541263, 533731692, 957921314

Offset: 1

Peter Luschny, Dec 05 2021

nonn

Column k=2 of A349802.
Cf. A000073, A057597 (tribonacci numbers for n <= 0), A006206 and A060280.
Cf. A349903, A349977.

read transforms;  # https://oeis.org/transforms.txt
arow := len -> EULERi([seq(A000073(n), n = 0..len)]): arow(39);
# second Maple program:
t:= n-> (<<0|1|0>, <0|0|1>, <1|1|1>>^n)[1, 3]:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; t(n-1)-b(n, n-1) end:
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 05 2021

(* EulerInvTransform is defined in A022562. *)
EulerInvTransform[LinearRecurrence[{1, 1, 1}, {0, 0, 1}, 40]]

InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n) = InvEulerT(Vec(x^2/(1 - x - x^2 - x^3) + O(x^n), -n)) \\ Andrew Howroyd, Dec 05 2021

# After the Maple program of Alois P. Heinz.
from functools import cache
from math import comb
def binomial(n, k):
    if n == -1: return 1
    return comb(n, k)
@cache
def A000073(n):
    if n <= 1: return 0
    if n == 2: return 1
    return A000073(n-1) + A000073(n-2) + A000073(n-3)
@cache
def b(n, i):
    if n == 0: return 1
    if i <  1: return 0
    return sum(binomial(a(i) + j - 1, j) *
               b(n - i * j, i - 1) for j in range(1 + n // i))
@cache
def a(n): return (A000073(n - 1) - b(n, n - 1))
print([a(n) for n in range(1, 41)])

def euler_invtrans(A) :
    L = []; M = []
    for i in range(len(A)) :
        s = (i+1)*A[i] - sum(L[j-1]*A[i-j] for j in (1..i))
        L.append(s)
        s = sum(moebius((i+1)/d)*L[d-1] for d in divisors(i+1))
        M.append(s/(i + 1))
    return M
@cached_function
def a(n): return a(n-1) + a(n-2) + a(n-3) if n > 2 else [0,0,1][n]
print(euler_invtrans([a(n) for n in range(40)]))

A022563 Number of 2-connected claw-free graphs on n nodes.

Original entry on oeis.org

0, 0, 1, 3, 8, 32, 126, 619, 3332, 20910, 157721, 1590330, 23074092, 487448150, 14466253437, 585505737038

Offset: 1

N. J. A. Sloane

R. Faudree, E. Flandrin and Z. Ryjacek, Claw-free graphs - a survey, Discr. Math., 164 (1997), 87-147.
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version e8699da.

Cf. A022562, A022564.

a(12) corrected using tinygraph by Falk Hüffner, Jan 18 2016

a(13)-a(16) from Brendan McKay, Jun 13 2021

A030268 Number of nonisomorphic connected partial lattices.

Original entry on oeis.org

1, 1, 1, 3, 9, 35, 153, 791, 4597, 29988, 215804, 1697291, 14457059, 132392971, 1295346365, 13468653637, 148142236784, 1716782858995, 20889118889021

Offset: 0

Christian G. Bower, revised Dec 28 2000

A partial lattice is a poset where every pair of points has a unique least upper (greatest lower) bound or has no upper (lower) bound.

N. J. A. Sloane, Transforms

A006966 = Cases[Import["https://oeis.org/A006966/b006966.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[Drop[A006966, 3]]] (* Jean-François Alcover, May 10 2019, updated Mar 17 2020 *)

Inverse Euler transform of A006966(n-2) (lattices).

a(17) (from A006966) from Jean-François Alcover, May 10 2019

a(18) (using A006966) from Alois P. Heinz, May 10 2019

A046745 Number of connected graphs with <= n edges.

Original entry on oeis.org

1, 2, 5, 10, 22, 52, 131, 358, 1068, 3390, 11461, 40964, 153786, 603927, 2471798, 10509270, 46296937, 210848414, 990794383, 4795761825, 23875074600, 122086530809, 640484152252, 3443478138871, 18954259427121, 106719731914780, 614115134054991, 3609008134177109

Offset: 1

N. J. A. Sloane

Andrew Howroyd, Table of n, a(n) for n = 1..60
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.

Partial sums of A002905.

A000664 = Cases[Import["https://oeis.org/A000664/b000664.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
A002905 = EulerInvTransform[Rest@A000664];
Accumulate[A002905] (* Jean-François Alcover, Jul 16 2022 *)

More terms from Vladeta Jovovic, Apr 10 2001
More terms from Vladeta Jovovic, Jul 05 2003
Terms a(25) and beyond added by Andrew Howroyd, May 06 2021

cp's OEIS Frontend

A005226 Number of atomic species of degree n; also number of connected permutation groups of degree n.

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A053465 Number of connected 2-multigraphs on n nodes.

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Programs

Mathematica

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A022564 Number of 2-connected non-Hamiltonian claw-free graphs on n nodes.

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A126757 Number of n-node connected graphs with no cycles of length less than 5.

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Author

Keywords

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Crossrefs

Programs

Mathematica

Formula

Extensions

A001928 Number of connected topologies with n unlabeled nodes.

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Programs

Mathematica

Formula

Extensions

A086991 Numbers of claw-free simple graphs (not necessarily connected) on n nodes.

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Extensions

A349904 Inverse Euler transform of the tribonacci numbers A000073.

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Maple

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PARI

Python

SageMath

A022563 Number of 2-connected claw-free graphs on n nodes.

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Links

Crossrefs

Extensions

A030268 Number of nonisomorphic connected partial lattices.

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