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. (Formerly M1153)

1, 2, 4, 8, 19, 53, 209, 1253, 13599, 288267, 12293435, 1031291299, 166122463891, 50668153831843, 29104823811067331, 31455590793615376099, 64032471295321173271027, 245999896624828253856990803, 1787823725042236528801735181651, 24639597076850046760911809226614419

Offset

0,2

Comments

a(n) is the number of isolated points over all simple unlabelled graphs with (n+1) nodes. - Geoffrey Critzer, Apr 14 2012

References

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..87

Formula

O.g.f.: A(x)/(1-x), where A(x) is o.g.f. for A000088. - Geoffrey Critzer, Apr 12 2012

a(n) = Sum_{k=0..n} A000088(k). - Petros Hadjicostas, Apr 19 2020

Example

a(2) = 4 includes the null graph G1 = [], G2 = [o], G3 = [o o], and G4 = [o-o].
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

Maple

b:= proc(n, i, l) `if`(n=0 or i=1, 1/n!*2^((p-> add(ceil((p[j]-1)/2)
      +add(igcd(p[k], p[j]), k=1..j-1), j=1..nops(p)))([l[], 1$n])),
       add(b(n-i*j, i-1, [l[], i$j])/j!/i^j, j=0..n/i))
    end:
a:= proc(n) option remember; b(n$2, [])+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2019

Mathematica

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 *)

Crossrefs

Partial sums of A000088.

Cf. A006896 (labeled case).

Sequence in context: A173310 A320178 A128816 * A287025 A034767 A005518

Adjacent sequences: A006894 A006895 A006896 * A006898 A006899 A006900

Keyword

easy,nonn,nice

Author

Colin Mallows

Extensions

Name edited by Petros Hadjicostas, Apr 08 2020

Status

approved