A003049 Number of connected Eulerian graphs with n unlabeled nodes.

1, 0, 1, 1, 4, 8, 37, 184, 1782, 31026, 1148626, 86539128, 12798435868, 3620169692289, 1940367005824561, 1965937435288738165, 3766548132138130650270, 13666503289976224080346733

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

These are connected graphs with every node of even degree (cf. A002854).

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 117.
• Valery A. Liskovets, Enumeration of Euler graphs. (Russian), Vesci Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk 1970, No.6, 38-46 (1970). Math. Rev., Vol. 44, 1972, p. 1195, #6557.
• R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969.
• R. W. Robinson, personal communication.
• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..88 (terms 1..60 from Max Alekseyev)
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Erich Friedman, Illustration of initial terms
• V. A. Liskovec, Enumeration of Euler Graphs, (in Russian), Akademiia Navuk BSSR, Minsk., 6 (1970), 38-46. (annotated scanned copy)
• C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880.
• C. L. Mallows and N. J. A. Sloane, Two-graphs, switching classes and Euler graphs are equal in number, SIAM J. Appl. Math., 28 (1975), 876-880. [Copy on N. J. A. Sloane's Home Page]
• Brendan McKay, Combinatorial Data (Eulerian graphs).
• R. W. Robinson, Enumeration of Euler graphs, pp. 147-153 of F. Harary, editor, Proof Techniques in Graph Theory. Academic Press, NY, 1969. (Annotated scanned copy)
• 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, Eulerian Graph.

Crossrefs

Cf. A002854, A008683.

Programs

Mathematica

A002854 = Import["https://oeis.org/A002854/b002854.txt", "Table"][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
EulerInvTransform[A002854] (* Jean-François Alcover, Aug 27 2019, updated Mar 17 2020 *)

Python

from functools import lru_cache
from itertools import combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A003049(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<>1)-1)*r+(q*r*(r-1)>>1) for q, r in p.items())+any(q&1 for q in p),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n)))
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(mobius(n//d)*c(d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 03 2024

Formula

Let B(x) = g.f. for A002854. Then g.f. A(x) for A003049 satisfies 1 + B(x) = exp(Sum_{n>=1} A(x^n)/n). - Robinson (1969).

Inverse Euler transform of A002854. (This is equivalent to the Robinson formula.) - Franklin T. Adams-Watters, Jul 24 2006

Let B(x) = g.f. for A002854. Then A(x) = Sum_{m >= 1} (mu(m)/m) * log(1 + B(x^m)), where mu(m) = A008683(m). (This is essentially a re-statement of the equation on p. 151 in Robinson (1969).) - Petros Hadjicostas, Feb 24 2021

Extensions

a(1)-a(26) were computed by R. W. Robinson

More terms from Vladeta Jovovic, Apr 18 2000