1,3

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 (qntmpkt(AT)yahoo.com), Jan 04 2008

S.-P. Eu, S.-C. Liu and Y.-N. Yeh, Odd or Even on Plane Trees, Discrete Math. 281 (2004), 189-196.

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*((1+4*x^2)^(1/2)-(1-4*x)^(1/2)-2*x)/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 19 2003

a(0)=0, 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

a(n)=sum{k=1..n, (1/n)*C(n,k)*C(n,k-1)*(1-(-1)^k)/2}; [From Paul Barry (pbarry(AT)wit.ie), Dec 16 2008]

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.

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 (deutsch(AT)duke.poly.edu), Feb 17 2007

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)]

a(n) + A071688 = A000108: Catalan numbers.

Cf. A001263.

Sequence in context: A018039 A084264 A088211 this_sequence A060816 A037552 A094618

Adjacent sequences: A071681 A071682 A071683 this_sequence A071685 A071686 A071687

easy,nonn

Sen-Peng Eu (giawgwan(AT)single.url.com.tw), Jun 23 2002

Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 25 2002