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A030982
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Number of noncrossing nonplanted bushes with n nodes, i.e., rooted noncrossing trees with n nodes and no nodes of degree 1.
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1
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0, 1, 1, 7, 18, 80, 284, 1169, 4628, 19137, 79165, 333058, 1410608, 6029816, 25941384, 112315945, 488862888, 2138161043, 9391903131, 41414729419, 183264846010, 813564012660, 3622193670040, 16170171489820, 72364908958800, 324586284275500, 1458976377988636
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OFFSET
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1,4
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to rooted trees
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FORMULA
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a(n) = 5*Sum_{k=2..n} ((-1)^(n-k)*2^(n-k)*k*C(n,k)*C(3*k-2,k-2)/(2*k+1))/n.
Recurrence: 2*n*(2*n+1)*a(n) = (n-1)*(11*n-12)*a(n-1) + 6*(9*n^2-21*n+8) * a(n-2) - 4*(n-3)*(11*n-56)*a(n-3) - 152*(n-4)*(n-3)*a(n-4). - Vaclav Kotesovec, Oct 24 2012
a(n) ~ 5*19^(n+1/2)/(27*sqrt(Pi)*4^(n+1)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012
a(n) = A030981(n) - A030980(n). - Andrew Howroyd, Nov 12 2017
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MATHEMATICA
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Table[5*Sum[(-1)^(n-k)*2^(n-k)*k*Binomial[n, k]*Binomial[3*k-2, k-2]/ (2*k+1), {k, 2, n}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Oct 24 2012 *)
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PROG
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(PARI) a(n) = 5*sum(k=2, n, (-1)^(n-k)*2^(n-k)*k*binomial(n, k)*binomial(3*k-2, k-2)/(2*k+1))/n; \\ Andrew Howroyd, Nov 12 2017
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CROSSREFS
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Cf. A030980, A030981.
Sequence in context: A019534 A024830 A262489 * A203381 A207158 A304992
Adjacent sequences: A030979 A030980 A030981 * A030983 A030984 A030985
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KEYWORD
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nonn,easy
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AUTHOR
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Emeric Deutsch
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STATUS
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approved
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