

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

R. W. Robinson, Table of n, a(n) for n = 1..26


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}];
Array[a, 18] (* JeanFrançois Alcover, Aug 29 2019, after Vladeta Jovovic *)


CROSSREFS

Cf. A158007, A000088, A002854.
Sequence in context: A222857 A108735 A143556 * A009576 A009580 A125839
Adjacent sequences: A007123 A007124 A007125 * A007127 A007128 A007129


KEYWORD

nonn


AUTHOR

N. J. A. Sloane.


STATUS

approved



