A003049 Number of connected Eulerian graphs with n unlabeled nodes. (Formerly M3344)

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

Offset

1,5

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.

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

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 *)

Prog

(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<<sum(p[r]*p[s]*gcd(r, s) for r, s in combinations(p.keys(), 2))+sum(((q>>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

Crossrefs

Cf. A002854, A008683.

Sequence in context: A370573 A335991 A063580 * A098563 A231398 A231465

Adjacent sequences: A003046 A003047 A003048 * A003050 A003051 A003052

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

More terms from Vladeta Jovovic, Apr 18 2000

Status

approved