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A030980
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Number of planted noncrossing bushes with n nodes; i.e., rooted noncrossing trees with n nodes, root degree 1 and no nonroot nodes of degree 1.
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1
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1, 0, 3, 4, 23, 66, 280, 1030, 4207, 16852, 69747, 289950, 1222540, 5192344, 22239672, 95864902, 415730735, 1812177000, 7936353049, 34901789568, 154067755503, 682428824890, 3032173906692, 13510960371744
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = Sum_{k=1..n} ((-1)^(n-k)*2^(n-k)*binomial(n, k)*binomial(3*k-2, k-1))/n.
G.f.: A(z) satisfies A(z)^3 + 2z*A(z)^3 - 2A(z)^2 - 4z*A(z)^2 + A(z) + 2z*A - z = 0.
D-finite with recurrence -2*n*(2*n-1)*a(n) +3*n*(n-2)*a(n-1) +30*(2*n-3)*(n-2)*a(n-2) +76*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 24 2022
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PROG
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(PARI) a(n) = sum(k=1, n, ((-1)^(n-k)*2^(n-k)*binomial(n, k)*binomial(3*k-2, k-1))/n) \\ Michel Marcus, Aug 03 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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