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A000226 Number of n-node unlabeled connected graphs with one cycle of length 3.
(Formerly M2668 N1066)
9
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, 182009003773, 513729375064, 1452007713130 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,3
COMMENTS
Number of rooted trees on n+1 nodes where root has degree 3. - Christian G. Bower
Third column of A033185. - Michael Somos, Aug 20 2018
From Washington Bomfim, Dec 22 2020: (Start)
Number of forests of 3 rooted trees with a total of n nodes.
Number of unicyclic graphs with a cycle of length 3 and a total of n nodes.
(End)
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
Alois P. Heinz, Table of n, a(n) for n = 3..1000 (first 198 terms from Vincenzo Librandi)
Sergei Abramovich, Partitions of integers, Ferrers-Young diagrams, and representational efficacy of spreadsheet modeling, Spreadsheets in Education (eJSiE): Vol. 5: Iss. 2, Article 1, p. 5
Eric Weisstein's World of Mathematics, Rooted Tree
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).
a(n) = Sum_{j1+2j2+···= n} (Product_{i=1..n} binomial(A000081(i) + j_i -1, j_i)) [(4.27) of [F. Ruskey] with n replaced by n+1]. - Washington Bomfim, Dec 22 2020
a(n) ~ (A187770 + A339986) * A051491^n / (2 * n^(3/2)). - Vaclav Kotesovec, Dec 25 2020
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..40); # Alois P. Heinz, Aug 21 2008
MATHEMATICA
terms = 30; r[_] = 0; Do[r[x_] = x *Exp[Sum[r[x^k]/k, {k, 1, j}]] + O[x]^j // Normal, {j, 1, terms+3}]; A[x_] = (r[x]^3 + 3*r[x]*r[x^2] + 2*r[x^3])/6 + O[x]^(terms+3); Drop[CoefficientList[A[x], x], 3] (* Jean-François Alcover, Nov 23 2011, updated Jan 11 2018 *)
PROG
(PARI) seq(max_n) = {my(a = f = vector(max_n), s, D); f[1]=1;
for(j=1, max_n - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1]));
for(n=3, max_n, s=0; forpart(P=n, D=Set(P); if(#D==3, s+=f[P[1]]*f[P[2]]*f[P[3]]; next());
if(#D==1, s+= binomial(f[P[1]]+2, 3); next());
if(P[1] == P[2], s += binomial(f[P[1]]+1, 2) * f[P[3]],
s += binomial(f[P[2]]+1, 2) * f[P[1]]), [1, n], [3, 3]); a[n] = s ); a[3..max_n] }; \\ Washington Bomfim, Dec 22 2020
CROSSREFS
Column 3 of A033185 and A217781.
For n >= 3 a(n) = A217781(n, 3) = A058879(n, n-2) = A033185(n, 3).
Sequence in context: A027967 A181306 A178035 * A291734 A291229 A036883
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 19 2000
STATUS
approved

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Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)