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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 (list; graph; refs; listen; history; text; internal format)
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 n-graphs into rooted Eulerian (n+1)-graphs: add an external root-vertex and connect it to the odd-valent 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(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. [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] (* Jean-François Alcover, Aug 29 2019, after Vladeta Jovovic *)
CROSSREFS
Sequence in context: A361729 A108735 A143556 * A009576 A009580 A125839
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 21 19:35 EDT 2024. Contains 372738 sequences. (Running on oeis4.)