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Search: id:A071688
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| A071688 |
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Number of plane trees with even number of leaves. |
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+0
4
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| 0, 1, 3, 7, 20, 66, 217, 715, 2424, 8398, 29414, 104006, 371384, 1337220, 4847637, 17678835, 64821680, 238819350, 883634026, 3282060210, 12233125112, 45741281820, 171529836218, 644952073662, 2430973096720, 9183676536076
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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S. P. Eu, S. C. Liu and Y. N. Yeh, Odd or Even on Plane Trees, 2002, Submitted
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FORMULA
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a(2n)= (1/(4*n+2))*binomial(4*n, 2*n), a(2n+1)= (1/(4*n+4))*binomial(4*n+2, 2*n+1)+(-1)^(n+1)*((1)/(2*n+2))*binomial(2*n, n)
G.f.: 1/4*(2-(1-4*x)^(1/2)+2*x-(1+4*x^2)^(1/2))/x. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Apr 19 2003
a(0)=1, a(n)=sum{k=0..floor(n/2), (1/n)*C(n,2k-1)C(n,2k)},n>0; - Paul Barry (pbarry(AT)wit.ie), Jan 25 2007
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EXAMPLE
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a(3) = 3 because among the 5 plane 3-trees there are 3 trees with even number of leaves; a(4) = 7 because among the 14 plane 4-trees there are 7 trees with even number of leaves.
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MATHEMATICA
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a[n_] := If[EvenQ[n], Binomial[2n, n]/(2n + 2), Binomial[2n, n]/(2n + 2) + (-1)^((n + 1)/2)Binomial[n - 1, (n - 1)/2]/(n + 1)]
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CROSSREFS
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a(n) + A071684 = A000108: Catalan numbers.
Adjacent sequences: A071685 A071686 A071687 this_sequence A071689 A071690 A071691
Sequence in context: A110490 A132868 A056783 this_sequence A110149 A024331 A007174
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KEYWORD
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easy,nonn
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AUTHOR
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Sen-Peng Eu (giawgwan(AT)single.url.com.tw), Jun 23 2002
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EXTENSIONS
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Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 25 2002
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