|
| |
|
|
A000226
|
|
Number of n-node unlabeled connected graphs with one cycle of length 3.
(Formerly M2668 N1066)
|
|
5
| |
|
|
1, 1, 3, 7, 18, 44, 117, 299, 793, 2095, 5607, 15047, 40708, 110499, 301541, 825784, 2270211, 6260800, 17319689, 48042494, 133606943, 372430476, 1040426154, 2912415527, 8167992598, 22947778342, 64577555147
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 3,3
|
|
|
COMMENTS
| Number of rooted trees where root has degree 3. - Christian Bower
|
|
|
REFERENCES
| J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 3..200
Index entries for sequences related to rooted trees
|
|
|
FORMULA
| G.f.: (r(x)^3+3*r(x)*r(x^2)+2*r(x^3))/6 where r(x) is g.f. for rooted trees (A000081).
|
|
|
MAPLE
| b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n, k) option remember; add(b(n+1-j*k), j=1..iquo(n, k)) end: B:= proc(n) option remember; unapply (add (b(k)*x^k, k=1..n), x) end: a:= n-> coeff (series ((B(n-2)(x)^3+ 3*B(n-2)(x)* B(n-2)(x^2)+ 2*B(n-2)(x^3))/6, x=0, n+1), x, n): seq (a(n), n=3..29); # Alois P. Heinz, Aug 21 2008
|
|
|
MATHEMATICA
| nmax = 29; b[n_] := (r[x_] := Sum[c[k]*x^k, {k, 0, n}]; c[0] = 0; cc = CoefficientList[ Series[r[x] - x*Exp[Sum[r[x^k]/k, {k, 1, n}]], {x, 0, n}], x]; solc = First[ Solve[ Thread[cc == 0]]]; gf[x_] := Sum[d[k]*x^k, {k, 0, n}]; dd = CoefficientList[ Series[ gf[x] - (r[x]^3 + 3*r[x]*r[x^2] + 2*r[x^3])/6 /. solc, {x, 0, n}], x]; sold = First[ Solve[ Thread[dd == 0]]]; Do[ If[c[k] =!= (ck = c[k] /. solc), c[k] = ck]; If[d[k] =!= (dk = d[k] /. sold), d[k] = dk], {k, 0, n}]); Do[ b[n], {n, 10, nmax + 10, 10}]; coes = CoefficientList[ gf[x] /. sold, x]; a[n_] := coes[[n+1]]; Table[a[n], {n, 3, nmax}] (* From Jean-François Alcover, Nov 23 2011 *)
|
|
|
CROSSREFS
| A001429. - W. Bomfim Feb, 10, 2011.
Sequence in context: A027967 A181306 A178035 * A036883 A191652 A191826
Adjacent sequences: A000223 A000224 A000225 * A000227 A000228 A000229
|
|
|
KEYWORD
| nonn,nice,changed
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 19 2000
|
| |
|
|
