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A061639
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Number of planar planted trees with n non-root nodes and every 2-valent node isolated.
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2
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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
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OFFSET
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0,5
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COMMENTS
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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. [From Paul Barry, Mar 07 2010]
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REFERENCES
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I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (Problem 2.7.4).
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LINKS
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Table of n, a(n) for n=0..27.
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FORMULA
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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))).
Contribution 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-4) +3*(4-n)*a(n-4)=0. - R. J. Mathar, Nov 15 2011
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CROSSREFS
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Sequence in context: A091468 A103457 A083587 * A008995 A111236 A164361
Adjacent sequences: A061636 A061637 A061638 * A061640 A061641 A061642
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KEYWORD
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nonn
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AUTHOR
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Vladeta Jovovic, Jun 13 2001
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STATUS
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approved
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