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A102893
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Number of noncrossing trees with n edges and having degree of the root at least 2.
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9
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1, 0, 1, 5, 25, 130, 700, 3876, 21945, 126500, 740025, 4382625, 26225628, 158331880, 963250600, 5899491640, 36345082425, 225082957512, 1400431689475, 8749779798375, 54874635255825, 345329274848250, 2179969531405680
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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a(0)=1; a(n) = 5*binomial(3n-1, n-2)/(3n-1) if n > 0.
G.f.: g - z*g^2, where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764).
D-finite with recurrence 2*n*(2*n+1)*(n-2)*a(n) -3*(n-1)*(3*n-4)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
a(n) ~ (5*3^(3*n + 1/2))/(36*4^n*n^(3/2)*sqrt(Pi)). - Peter Luschny, Aug 08 2020
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EXAMPLE
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a(2)=1 because among the noncrossing trees with 2 edges, namely /_, _\ and /\, only the last one has root degree >1.
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MAPLE
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a:=proc(n) if n=0 then 1 else 5*binomial(3*n-1, n-2)/(3*n-1) fi end:
seq(a(n), n=0..25);
# Recurrence:
a := proc(n) option remember; if n < 3 then return [1, 0, 1][n+1] fi;
(27*n^3 - 81*n^2 + 78*n - 24)*a(n - 1)/(4*n^3 - 6*n^2 - 4*n) end:
alias(PS=ListTools:-PartialSums): A102893List := proc(m) local A, P, n;
A := [1, 0]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-2]] od; A end: A102893List(23); # Peter Luschny, Mar 26 2022
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MATHEMATICA
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a[0] = 1; a[n_] := 5*Binomial[3n-1, n-2]/(3n-1); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 01 2018 *)
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PROG
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(PARI) a(n) = if(n<=1, n==0, 5*binomial(3*n-1, n-2)/(3*n-1)); \\ Andrew Howroyd, Nov 17 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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