1,3

a(n)= # of ways to dissect convex (n+1)-gon with non-crossing diagonals so that no 2m-gons (m>1) appear - Len Smiley (smiley(AT)math.uaa.alaska.edu)

Number of even trees (i.e. ordered trees in which all nodes have even outdegree) with n leaves. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 06 2002

a(n) = # permutations on [n-1] in which the last 2 entries of each 321 pattern are adjacent in position. For example, a(5)=20 counts all permutations on [4] except 3241, 4231, 4312, 4321, the first, for instance because the 2 and 1 are not adjacent. - David Callan (callan(AT)stat.wisc.edu), Jul 20 2005

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

Index entries for reversions of series

L. Smiley, Even-gon reference

L. Smiley, [math/9907057] Variants of Schroeder Dissections

G.f. satisfies: A(x) = x + A(x)^2/(1-A(x)^2); by Lagrange Inversion: A(x) = x + sum_{n>=0} d^n/dx^n (x^2/(1-x^2))^(n+1)/(n+1)!, or: A(x) = sum_{n>=0} sum_{k>=n} C(k-1, k-n)*(2k)!/(2k-n+1)!*x^(2k-n+1)/n!. - Paul D. Hanna (pauldhanna(AT)juno.com), Mar 24 2004

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(2n-2k, n)/(n+1) for n>0, with a(0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2004

a(3)=2 because one diagonal may be placed 2 ways in the quadrilateral (placing none is not allowed).

Generated from Fibonacci polynomials (A011973) and odd self-convolutions of Catalan numbers (A039599):

a(1) = 1*1

a(2) = 1*2 + 0*1/3

a(3) = 1*5 + 1*3/3

a(4) = 1*14 + 2*9/3 + 0*1/5

a(5) = 1*42 + 3*28/3 + 1*5/5

a(6) = 1*132 + 4*90/3 + 3*20/5 + 0*1/7

a(7) = 1*429 + 5*297/3 + 6*75/5 + 1*7/7

a(8) = 1*1430 + 6*1001/3 + 10*275/5 + 4*35/7 + 0*1/9

This process is equivalent to the formula:

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,n-2k-1)*C(2n-2k,n-2k)/(n+1).

The odd self-convolutions of Catalan numbers begin:

A000108^1: {1,1,2,5,14,42,132,329,1430,...}

A000108^3: {1,3,9,28,90,297,1001,...}

A000108^5: {1,5,20,75,275,...}

A000108^7: {1,7,35,...}

Order := 20; solve(series((A-A^2-A^3)/(1-A^2), A)=x, A);

(PARI) {a(n)=polcoeff(sum(m=0, n, sum(k=0, n, binomial(k+m-1, k)*binomial(2*k+2*m, m)*x^(2*k+m+1)/(2*k+m+1))), n)}

(PARI) {a(n)=if(n==0, 1, sum(k=0, (n-1)\2, binomial(n-k-1, k)*binomial(2*n-2*k, n))/(n+1))} (Hanna)

Cf. A003168.

Cf. A000108.

Sequence in context: A108600 A128729 A006027 this_sequence A049141 A049129 A063376

Adjacent sequences: A049121 A049122 A049123 this_sequence A049125 A049126 A049127

nonn,easy,nice

Olivier Gerard (ogerard(AT)ext.jussieu.fr)

More terms from Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2004