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 A007857 Number of independent sets in rooted plane trees on n nodes. 4
 1, 2, 8, 37, 184, 959, 5172, 28641, 162008, 932503, 5445934, 32197334, 192357788, 1159603592, 7045356104, 43098733353, 265240985112, 1641100253735, 10202295895890, 63696629668980, 399216722146770, 2510833297584165 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equals the main diagonal of square array A130523. - Paul D. Hanna, Jun 06 2007 LINKS M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics, Vol. 18, No. 2 (1997), 195-210. M. Klazar, Addendum Twelve Countings with Rooted Plane Trees, European Journal of Combinatorics, Vol. 18, No. 6 (1997), 739-740. FORMULA a(n+1) = (2/(n+1))*C(3n,n)-(1/(n+1))*C(2n,n) = A007226(n)-A000108(n). - Paul Barry, Nov 05 2006 G.f.: A(x) = x/[1 - xC(x)F(x) - xF(x)^2] where C(x) is g.f. of Catalan numbers (A000108): C(x) = 1 + xC(x)^2 and F(x) is g.f. of ternary numbers (A001764): F(x) = 1 + xF(x)^3. - Paul D. Hanna, Jun 06 2007 Conjecture: 2*n*(n-1)*(2*n-3)*(44*n-69)*a(n) +(n-1)*(176*n^3-9591*n^2+38703*n-40640)*a(n-1) +(-17479*n^4+218005*n^3-959616*n^2+1797890*n-1221920)*a(n-2) +6*(3*n-10)*(2*n-7)*(3*n-11)*(517*n-1198)*a(n-3)=0. - R. J. Mathar, Nov 26 2012 PROG (PARI) {a(n)=my(A000108, A001764); A000108=Ser(vector(n+1, r, binomial(2*r-2, r-1)/r)); A001764=Ser(vector(n+1, r, binomial(3*r-3, r-1)/(2*r-1))); polcoeff(x/(1-x*A000108*A001764-x*A001764^2 +x*O(x^n)), n)} \\ Paul D. Hanna, Jun 06 2007 CROSSREFS Cf. A000108, A001764; A130523. Sequence in context: A224032 A046814 A305547 * A289541 A047729 A020076 Adjacent sequences:  A007854 A007855 A007856 * A007858 A007859 A007860 KEYWORD nonn AUTHOR EXTENSIONS More terms from Paul Barry, Nov 05 2006 STATUS approved

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Last modified April 26 12:00 EDT 2019. Contains 322472 sequences. (Running on oeis4.)