

A007126


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


2



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

1,5


COMMENTS

Comment from Valery Liskovets. Mar 13 2009: 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 ngraphs into rooted Eulerian (n+1)graphs: add an external rootvertex and connect it to the oddvalent vertices.


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



FORMULA

Comment from Vladeta Jovovic, Mar 15 2009: It is not difficult to prove that a(n) = A000088(n1)  Sum_{k=1..n1} a(k)*A002854(nk), 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. [Vladeta Jovovic, Mar 14 2009]


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}];


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



