3,3

Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Disc. Math., 204 (1999) 73- (as given in A005043)

a(n,k) = Binomial[n,k]Binomial[n-2-k,k]/(k+1). - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 22 2008: (Start)

O.g.f. : 1 + x + sqrt(1 - 2*x + x^2*(1 - 4*a))]/(2*x*(1 + a*x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... .

Define a functional I on formal power series of the form f(x) = 1 + a*x + b*x^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).

Let now f(x) = 1 + a*x^2 + a*x^3 + a*x^4 + ... . Then the o.g.f. for this table is I(f(x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . Cf. A001263 and A108767. (End)

A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.

Triangle begins:

1,

1,2,

1,5,

1,9,5,

1,14,21,

1,20,56,14,

1,27,120,84,

1,35,225,300,42,

1,44,385,825,330, ...

Sequence in context: A064865 A093127 A115123 this_sequence A054251 A163963 A119763

Adjacent sequences: A132078 A132079 A132080 this_sequence A132082 A132083 A132084

nonn,tabf

Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 01 2008 at the suggestion of R. J. Mathar