|
%I #39 Feb 06 2022 06:57:57
%S 1,1,2,7,22,66,212,715,2438,8398,29372,104006,371516,1337220,4847208,
%T 17678835,64823110,238819350,883629164,3282060210,12233141908,
%U 45741281820,171529777432,644952073662,2430973304732,9183676536076
%N Number of plane trees with n edges and having an odd number of leaves.
%C Narayana transform (A001263) of [1, 0, 1, 0, 1, 0, 1, ...]. Example: a(4) = 7 = (1, 6, 6, 1) dot (1, 0, 1, 0) = (1 + 0 + 6 + 0). - _Gary W. Adamson_, Jan 04 2008
%H Robert Israel, <a href="/A071684/b071684.txt">Table of n, a(n) for n = 1..1668</a>
%H Yu Hin Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv:1912.00555 [math.CO], 2019.
%H S. P. Eu, S. C. Liu and Y. N. Yeh, <a href="https://doi.org/10.1016/j.disc.2003.07.011">Odd or Even on Plane Trees</a>, Discrete Math. 281 (2004), 189-196.
%F a(2*n) = (1/(4*n + 2))*binomial(4*n, 2*n);
%F a(2*n-1) = (1/(4*n))*binomial(4*n-2, 2*n-1) - (-1)^n*(1/(2*n))*binomial(2*n-2, n-1), with n>0.
%F G.f.: (1/4)*((1+4*x^2)^(1/2) - (1-4*x)^(1/2)-2*x)/x. - _Vladeta Jovovic_, Apr 19 2003
%F a(0)=0; a(n) = Sum_{k = 0..floor(n/2)} (1/n)*C(n,2*k+1)*C(n,2*k) for n>0. - _Paul Barry_, Jan 25 2007
%F a(n) = Sum_{k=1..n} (1/n)*C(n,k)*C(n,k-1)*(1-(-1)^k)/2. - _Paul Barry_, Dec 16 2008
%F Conjecture: n*(n+1)*(10*n-37)*a(n) + 2*n*(5*n^2-42*n+91)*a(n-1) + 4*(-40*n^3+270*n^2-560*n+357)*a(n-2) + 8*(n-3)*(5*n^2-42*n+91)*a(n-3) - 16*(n-4)*(25*n-51)*(2*n-7)*a(n-4) = 0. - _R. J. Mathar_, Jul 05 2018
%F a(n) = (A000108(n) + 2^n * binomial(1/2, (n+1)/2))/2. - _Vladimir Reshetnikov_, Oct 03 2016
%F 32*n*(2*n+1)*a(n) - 48*(n+2)*(n+1)*a(n+1) + 8*(n^2-n-9)*a(n+2) - 4*(2*n^2+10*n+9)*a(n+3) - 2*(n+5)*(n+6)*a(n+4) + (n+5)*(n+6)*a(n+5) = 0. - _Robert Israel_, Jul 05 2018
%e a(3)=2 because among the 5 plane 3-trees there are 2 trees with odd number of leaves; a(4)=7 because among the 14 plane 4-trees there are 7 trees with odd number of leaves.
%p G:=((1+4*x^2)^(1/2)-(1-4*x)^(1/2)-2*x)/4/x: Gser:=series(G,x=0,30): seq(coeff(Gser,x,n),n=1..26); # _Emeric Deutsch_, Feb 17 2007
%t 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)]
%t Table[(CatalanNumber[n] + 2^n Binomial[1/2, (n + 1)/2])/2, {n, 20}] (* _Vladimir Reshetnikov_, Oct 03 2016 *)
%Y a(n) + A071688 = A000108: Catalan numbers.
%Y Cf. A001263, A007595.
%K nonn,easy
%O 1,3
%A _Sen-peng Eu_, Jun 23 2002
%E Edited by _Robert G. Wilson v_, Jun 25 2002
|
