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A071688 Number of plane trees with even number of leaves. 4
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; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

S. P. Eu, S. C. Liu and Y. N. Yeh, Odd or Even on Plane Trees, 2002, Submitted

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

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, 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, Jan 25 2007

a(n) = 0^n + Sum{k=1..n} (1/n)*C(n,k)*C(n,k-1)*(1+(-1)^k)/2. - Paul Barry, Dec 16 2008

a(n) = Sum_{k=0..n} ( Sum_{j=0..n-k} ( (-1)^j*(C(n,2k)*C(n,2k+j) - C(n,2k-1)*C(n,2k+j+1) )). - Paul Barry, Sep 13 2010

Conjecture: (n+2)*(n+1)*a(n) -2*(n+2)*(n+1)*a(n-1) +4*(-2*n^2+6*n-1)*a(n-2) +8*(n^2-9*n+11)*a(n-3) 48*(n-2)*(n-3)*a(n-4) +32*(2*n-7)*(n-4)*a(n-5)=0. - R. J. Mathar, Nov 24 2012

a(n) = (A000108(n) - 2^n * binomial(1/2, (n+1)/2))/2. - Vladimir Reshetnikov, Oct 03 2016

From Vaclav Kotesovec, Oct 04 2016: (Start)

Recurrence (of order 3): n*(n+1)*(5*n^2 - 20*n + 18)*a(n) = 2*n*(2*n - 5)*(5*n^2 - 10*n + 3)*a(n-1) - 4*(n-2)*n*(5*n^2 - 20*n + 18)*a(n-2) + 8*(n-3)*(2*n - 5)*(5*n^2 - 10*n + 3)*a(n-3).

a(n) ~ 2^(2*n-1)/(sqrt(Pi*n)*n).

(End)

a(n) = A119358(n) - A119359(n) = hypergeom([1/2-n/2, 1/2-n/2, -n/2, -n/2], [1/2, 1/2, 1], 1) - hypergeom([-1/2-n/2, 1/2-n/2, 1-n/2, -n/2], [1/2, 1/2, 1], 1]. - Vladimir Reshetnikov, Oct 05 2016

EXAMPLE

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.

MATHEMATICA

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)]

Table[(CatalanNumber[n] - 2^n Binomial[1/2, (n + 1)/2])/2, {n, 20}] (* Vladimir Reshetnikov, Oct 03 2016 *)

CROSSREFS

a(n) + A071684 = A000108: Catalan numbers.

Cf. A007595.

Sequence in context: A320740 A320741 A292503 * A232687 A211602 A110149

Adjacent sequences:  A071685 A071686 A071687 * A071689 A071690 A071691

KEYWORD

easy,nonn

AUTHOR

Sen-peng Eu, Jun 23 2002

EXTENSIONS

Edited by Robert G. Wilson v, Jun 25 2002

STATUS

approved

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Last modified April 24 16:06 EDT 2019. Contains 322430 sequences. (Running on oeis4.)