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A006897 a(n) is the number of hierarchical linear models on n unlabeled factors allowing 2-way interactions (but no higher order interactions); or the number of unlabeled simple graphs with <= n nodes.

%I A006897 M1153 #52 Sep 29 2024 11:56:02
%S A006897 1,2,4,8,19,53,209,1253,13599,288267,12293435,1031291299,166122463891,
%T A006897 50668153831843,29104823811067331,31455590793615376099,
%U A006897 64032471295321173271027,245999896624828253856990803,1787823725042236528801735181651,24639597076850046760911809226614419
%N A006897 a(n) is the number of hierarchical linear models on n unlabeled factors allowing 2-way interactions (but no higher order interactions); or the number of unlabeled simple graphs with <= n nodes.
%C A006897 a(n) is the number of isolated points over all simple unlabeled graphs with (n+1) nodes. - _Geoffrey Critzer_, Apr 14 2012
%D A006897 R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
%D A006897 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006897 Alois P. Heinz, <a href="/A006897/b006897.txt">Table of n, a(n) for n = 0..87</a>
%H A006897 Max A. Alekseyev and Allan Bickle, <a href="https://allanbickle.wordpress.com/wp-content/uploads/2016/05/forbidsubfinal.pdf">Forbidden Subgraphs of Single Graphs</a>, (2024). See p. 13.
%F A006897 O.g.f.: A(x)/(1-x), where A(x) is o.g.f. for A000088. - _Geoffrey Critzer_, Apr 12 2012
%F A006897 a(n) = Sum_{k=0..n} A000088(k). - _Petros Hadjicostas_, Apr 19 2020
%e A006897 a(2) = 4 includes the null graph G1 = [], G2 = [o], G3 = [o o], and G4 = [o-o].
%e A006897 a(3) = 8 includes the null graph G1 = [], G2 = [o], G3 = [o o], G4 = [o-o], G5 = [o o o], G6 = [o-o o], G7 = [o-o-o], and G8 = [triangle with three unlabeled nodes]. - _Petros Hadjicostas_, Apr 10 2020
%p A006897 b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
%p A006897       +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
%p A006897        add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
%p A006897     end:
%p A006897 a:= proc(n) option remember; b(n$2, [])+`if`(n>0, a(n-1), 0) end:
%p A006897 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 14 2019
%t A006897 nn = 15; g = Sum[NumberOfGraphs[n] x^n, {n, 0, nn}]; CoefficientList[Series[g/(1 - x), {x, 0, nn}], x]  (* _Geoffrey Critzer_, Apr 12 2012 *)
%Y A006897 Partial sums of A000088.
%Y A006897 Cf. A006896 (labeled case).
%K A006897 easy,nonn,nice
%O A006897 0,2
%A A006897 _Colin Mallows_
%E A006897 Name edited by _Petros Hadjicostas_, Apr 08 2020