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A007126 Number of connected rooted strength 1 Eulerian graphs with n nodes.

%I A007126 M4126 #25 Jul 08 2025 17:00:22
%S A007126 1,0,1,1,6,18,111,839,11076,260327,11698115,1005829079,163985322983,
%T A007126 50324128516939,29000032348355991,31395491269119883535,
%U A007126 63967623226983806252862,245868096558697545918087280
%N A007126 Number of connected rooted strength 1 Eulerian graphs with n nodes.
%C A007126 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.
%D A007126 R. W. Robinson, personal communication.
%D A007126 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.
%D A007126 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007126 Chai Wah Wu, <a href="/A007126/b007126.txt">Table of n, a(n) for n = 1..88</a> (terms 1..26 from R. W. Robinson)
%F A007126 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).
%F A007126 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]
%t A007126 A000088 = Cases[Import["https://oeis.org/A000088/b000088.txt", "Table"], {_, _}][[All, 2]];
%t A007126 A002854 = Import["https://oeis.org/A002854/b002854.txt", "Table"][[All, 2]];
%t A007126 a[n_] := a[n] = A000088[[n]] - Sum[a[k] A002854[[n - k]], {k, 1, n - 1}];
%t A007126 Array[a, 18] (* _Jean-François Alcover_, Aug 29 2019, after _Vladeta Jovovic_ *)
%Y A007126 Cf. A158007, A000088, A002854.
%K A007126 nonn
%O A007126 1,5
%A A007126 _N. J. A. Sloane_