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 A061639 Number of planar planted trees with n non-root nodes and every 2-valent node isolated. 3
 0, 1, 1, 1, 4, 10, 28, 85, 262, 829, 2677, 8776, 29143, 97825, 331381, 1131409, 3889381, 13450744, 46764532, 163357807, 573064849, 2018027719, 7131064045, 25278463756, 89866690732, 320328538033, 1144591699069, 4099050204445, 14710315969696, 52893571881142 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Isolated 2-valent node is a 2-valent node non-adjacent to any other 2-valent node. Hankel transform of a(n+1) is A174108. Binomial transform of a(n+1) is A174107. - Paul Barry, Mar 07 2010 REFERENCES I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.7.4). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{m=0..n-1} Sum_{i=0..n-m-1} (-1)^i/(m+1)*binomial(2*m, m)*binomial(m+i, i)*binomial(m+i+1, n-m-i-1). G.f.: 1/2*(1-sqrt(1-4*(x+x^2)/(1+x+x^2))). From Paul Barry, Mar 07 2010: (Start) G.f.: (x(1+x)/(1+x+x^2))c(x(1+x)/(1+x+x^2)), c(x) the g.f. of A000108; G.f.: (1+x+x^2-sqrt(1-2x-5x^2-6x^3-3x^4))/(2*(1+x+x^2)). (End) Conjecture: n*a(n) +2*(2-n)*a(n-1) +(14-5*n)*a(n-2) +6*(3-n)*a(n-3) +3*(4-n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011 a(n) ~ 7^(1/4) * 2^(n-7/2) * 3^(n+1/4) / (sqrt(Pi) * n^(3/2) * (sqrt(21)-3)^(n-1/2)). - Vaclav Kotesovec, Feb 12 2014 From Peter Bala, May 30 2017: (Start) G.f. A(x) satisfies the differential equation (3*x^4 + 6*x^3 + 5*x^2 + 2*x - 1)*A(x)' - (4*x + 2)*A(x) + 2*x + 1 = 0 with A(0) = 0. Mathar's conjectural recurrence above follows from this. 1 - A(x) + A(x)^2 = 1/(1 + x + x^2). (End) MATHEMATICA CoefficientList[Series[1/2*(1-Sqrt[1-4*(x+x^2)/(1+x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *) CROSSREFS Sequence in context: A103457 A083587 A228403 * A243600 A239577 A008995 Adjacent sequences:  A061636 A061637 A061638 * A061640 A061641 A061642 KEYWORD nonn,easy AUTHOR Vladeta Jovovic, Jun 13 2001 STATUS approved

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Last modified August 8 06:27 EDT 2020. Contains 336290 sequences. (Running on oeis4.)