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 A000226 Number of n-node unlabeled connected graphs with one cycle of length 3. (Formerly M2668 N1066) 8
 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 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 Sergei Abramovich, Spreadsheets in Education (eJSiE): Vol. 5: Iss. 2, Article 1, p. 5 Eric Weisstein's World of Mathematics, Frequency Representation 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_(P) { C(f(p1)+a1-1, a1) * C(f(p2)+a2-1, a2) * C(f(p3)+a3-1, a3) }, where P is a partition of n, (p1^a1 p2^a2 p3^a3 ...); f(n) = A000081(n), n >= 1, and C(,) is a binomial coefficient. - Washington Bomfim, Jul 06 2012 EXAMPLE a(7) = 18 because the partitions of 7 correspond respectively, (1^2 5) => binomial(f(1)+2-1, 2) * f(5) = 9, (1 2 4) => f(1) * f(2) * f(4) = 4, (1 3^2) => f(1) * binomial(f(3)+2-1, 2) = 3, (2^2 3) => binomial(f(2)+2-1, 2) * f(3) = 2; and 9+4+3+2 = 18. 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) f = vector(200);                             \\ f[n] = A000081[n], n=1..200 sum2(k) = {local(s); s=0; fordiv(k, d, s += d * f[d]); return(s)}; Init_f() = {f[1]= 1; for(n=1, 199, s=0; for(k=1, n, s += sum2(k)*f[n-k+1]); f[n+1]=s/n)}; visit(p1, p2, p3) = {                       \\ Visit partition p1, p2, p3 if((p1

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Last modified March 18 15:05 EDT 2018. Contains 300771 sequences. (Running on oeis4.)