The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A216785 Number of unlabeled graphs on n nodes that have exactly two non-isomorphic components. 11
0, 0, 1, 2, 8, 28, 145, 1022, 12320, 274785, 12007355, 1019030127, 165091859656, 50502058491413, 29054157815353374, 31426486309136268658, 64001015806929213894372 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Stated more precisely: Number of unlabeled graphs on n nodes that have exactly two connected components and these components are not isomorphic (and nonempty).
REFERENCES
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 48.
LINKS
FORMULA
O.g.f.: A(x)^2/2 - A(x^2)/2 where A(x) is the o.g.f. for A001349 after setting A001349(0)=0.
EXAMPLE
a(4)=2 = 1*2 where 1*2=A001349(1)*A001349(3) counts graphs with a component of 1 node and a component with 3 nodes. There is no contribution with a component of 2 nodes and another component of 2 nodes because A001349(2)=1 means these components would be isomorphic. - R. J. Mathar, Jul 18 2016
a(5)=8 = 1*6 + 1*2 where 1*6=A001349(1)*A001349(4) counts graphs with a component of 1 node and a component with 4 nodes, and where 1*2 = A001349(2)*A001349(3) counts graphs with a component of 2 nodes and a component of 3 nodes. - R. J. Mathar, Jul 18 2016
MATHEMATICA
Needs["Combinatorica`"]; max=25; A000088=Table[NumberOfGraphs[n], {n, 0, max}]; f[x_]=1-Product[1/(1-x^k)^a[k], {k, 1, max}]; a[0]=a[1]=a[2]=1; coes=CoefficientList[Series[f[x], {x, 0, max}], x]; sol=First[Solve[Thread[Rest[coes+A000088]==0]]]; cg=Table[a[n], {n, 1, max}]/.sol; Take[CoefficientList[CycleIndex[AlternatingGroup[2], s]-CycleIndex[SymmetricGroup[2], s]/.Table[s[j]->Table[Sum[cg[[i]] x^(k*i), {i, 1, max}], {k, 1, max}][[j]], {j, 1, 3}], x], {4, max}] (* after code given by Jean-François Alcover in A001349 *)
PROG
(PARI) {c=[1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476, 50335907869219, 29003487462848061, 31397381142761241960, 63969560113225176176277, 245871831682084026519528568, 1787331725248899088890200576580, 24636021429399867655322650759681644]; }
for(n=3, 19, print([n, sum(j=1, (n-1)\2, c[j]*c[n-j])+if(n%2==0, c[n/2]*(c[n/2]-1)/2)]
)); /* David Broadhurst, Jul 18 2016 */
CROSSREFS
Cf. A058915, A001349, A217955, A275165, A275166 (allows an empty component), A274934 (allows isomorphic components).
Sequence in context: A150732 A225689 A330211 * A261559 A061230 A241627
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Oct 15 2012
EXTENSIONS
Two zeros prepended (offset changed), formula updated, and entries corrected by R. J. Mathar, N. J. A. Sloane, Jul 18 2016. (Thanks to Allan C. Wechsler for pointing out that all the entries above a(19) were wrong.)
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 12 16:34 EDT 2024. Contains 373334 sequences. (Running on oeis4.)