A007126 - OEIS

A007126

Number of connected rooted strength 1 Eulerian graphs with n nodes.
(Formerly M4126)

1, 0, 1, 1, 6, 18, 111, 839, 11076, 260327, 11698115, 1005829079, 163985322983, 50324128516939, 29000032348355991, 31395491269119883535, 63967623226983806252862, 245868096558697545918087280, 1787318122744773653459580586721, 24635927579960347335247422099617908

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

Here strength 1 means that the graph is a simple graph (i.e. without multiple edges and loops). Cf. the description of A002854 (number of Euler graphs); and the initial terms 1, 0, 1, 1, 6 can be easily verified. By the way, there is a simple bijective transformation of arbitrary n-graphs into rooted Eulerian (n+1)-graphs: add an external root-vertex and connect it to the odd-valent vertices. - Valery Liskovets, Mar 13 2009

REFERENCES

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..26 from R. W. Robinson)

FORMULA

From Vladeta Jovovic, Mar 15 2009: (Start)

It is not difficult to prove that a(n) = A000088(n-1) - Sum_{k=1..n-1} a(k)*A002854(n-k), n>1, with a(1) = 1, which is equivalent to the conjecture that the Euler transform of A158007(n) gives A007126(n+1) (see A158007).

O.g.f.: x*G(x)/(1+H(x)), where G(x) = 1+x+2*x^2+4*x^3+11*x^4+34*x^5+... = o.g.f for A000088 and H(x) = x+x^2+2*x^3+3*x^4+7*x^5+16*x^6+... = o.g.f for A002854. (End)

MATHEMATICA

A000088 = Cases[Import["https://oeis.org/A000088/b000088.txt", "Table"], {_, _}][[All, 2]];

A002854 = Import["https://oeis.org/A002854/b002854.txt", "Table"][[All, 2]];

a[n_] := a[n] = A000088[[n]] - Sum[a[k] A002854[[n - k]], {k, 1, n - 1}];

Array[a, 18] (* Jean-François Alcover, Aug 29 2019, after Vladeta Jovovic *)

CROSSREFS

Cf. A158007, A000088, A002854.

Sequence in context: A361729 A108735 A143556 * A009576 A009580 A125839

Adjacent sequences: A007123 A007124 A007125 * A007127 A007128 A007129

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved