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

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Last modified October 2 19:38 EDT 2022. Contains 357228 sequences. (Running on oeis4.)