Some relatives of the Catalan sequence

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Abstract

We study a family of sequences cn(a2,,ar), where r2 and a2,,ar are real parameters. We find a sufficient condition for positive definiteness of the sequence cn(a2,,ar) and check several examples from OEIS. We also study relations of these sequences with the free and monotone convolution.

Introduction

The Catalan sequence (2n+1n)12n+1:1,1,2,5,14,42,132,429,1430,4862,16796, (entry A000108 in OEIS [19]) plays an important role in mathematics. It counts several combinatorial objects, such as binary trees, noncrossing partitions, Dyck paths, dissections of a polygon into triangles and many others, as can be seen in the Stanley's book [20]. It also shows up as the moment sequence of the Marchenko-Pastur law and (in its aerated version) of the Wigner law, which in the free probability theory are the analogs of the Poisson and the normal law.

One of the generalizations are Fuss numbers (pn+1n)1pn+1, which count for example p-ary trees with npn+1 leaves, see [6]. The generating functionBp(z):=n=0(np+1n)znnp+1 satisfies equalityBp(z)=1+zBp(z)p. Lambert observed that the powers of Bp(z) admit quite nice Taylor expansion:Bp(z)r=n=0(np+rn)rznnp+r. The coefficients (np+rn)rnp+r have also combinatorial interpretations, see [6], and are called generalized Fuss numbers or Raney numbers. Formulas (2), (3) remain true if the parameters p,r are real.

It turns out that for p,rR the sequence (np+rn)rnp+r is positive definite if and only if either p1,0<rp or p0,p1r<0 or r=0, see [11], [12], [13], [5], [8]. Moreover, the corresponding probability distributions μ(p,r) are interesting from the point of view of noncommutative probability, for example the free R-transform of μ(p,r) is Bpr(z)r1, μ(1+p,1)=μ(1,1)p, p>0, where “⊠” is the multiplicative free convolution, and μ(p,q)μ(p+r,r)=μ(p+r,q+r), where “▷” is the monotone convolution, see [14].

In this paper we are going to study sequences cn(a) which are defined by real parameters a=(a2,,ar) and recurrence relation (4), with c0(a)=1. If r=2, a2=1 then cn(a) is the Catalan sequence. First we provide combinatorial motivations for such sequences, for example in terms of action of a finite family of operators on a product. Formula (4) implies equations (6) and (7) for the generating and the upshifted generating function Ca(z), Da(z):=zCa(z).

In Section 2 we study these functions, in particular we give a sufficient condition when the domain of Da(z) and Ca(z) is contained in CR. Consequently, Da is a Pick function and then the sequence cn(a) is positive definite. If this is the case then the corresponding probability distribution will be denoted μ(a). In Section 3 we find the free R-transform for μ(a) which is a rational function, and study the convolution semigroup μ(a)t. We also prove that if μ(a) is freely infinitely divisible then r3. Next we observe that the considered class of measures is closed under monotone convolution: μ(a)μ(a)=μ(a), where the sequence a is given by (18).

In Section 6 we study in details the case r=3. We give formulas for cn(a,b), free cumulants κn(a,b), the generating functions Ca,b(z),Da,b(z), and prove that the sequence cn(a,b) is positive definite if and only if a2+3b0. Moreover, we show that for a,b>0 the distribution μ(a,b) is freely infinitely divisible. For the case r=4 we provide sufficient conditions for positive definiteness (Theorem 7.2, Proposition 7.3). Then we consider the symmetric case, i.e. when aj=0 whenever j is even.

In the final section we provide a record of several integer sequences from OEIS which are of the form cn(a) and verify their positive definiteness.

Section snippets

Operators on a set

In this section we assume that a:=(a2,,ar), r2, is a fixed finite sequence of positive integers. Suppose that X is a set and that for j=2,3,,r we are given some j-ary operators on X, Fi(j):XjX, i=1,2,,aj. Denote by cn(a) the number of all possible compositions of these operations applied to the product x0x1xn. For example, for n=2 we can apply eitherFi(3)(x0,x1,x2)orFi1(2)(Fi2(2)(x0,x1),x2)orFi1(2)(x0,Fi2(2)(x1,x2)), where 1ia3, 1i1,i2a2, so that c2(a)=a3+2a22. Each such composed

The generating functions and positive definiteness

Motivated by the previous section we are going to study sequences defined by the recurrence relation (4), with c0(a):=1, where a=(a2,,ar)Rr1, r2, with ar0. The generating function Ca(z), given by (5), and the upshifted generating function Da(z):=zCa(z), satisfy relations (6) and (7), with Da(0)=0.

First we note the following homogeneity:

Proposition 2.1

If d0 and b=(da2,d2a3,,dr1ar) thendncn(a)=cn(b),Ca(dz)=Cb(z),Da(dz)=dDb(z).

Proof

This is a direct consequence of (4) and (5). 

Proposition 2.2

The functions Ca(z), Da(z)

R-transform and free additive convolution semigroups

For a sequence 1=c0,c1,c2, with generating function C(z):=n=0cnzn (possibly a formal power series), the free R-transform of cn, or of C(z), is defined by1+R(zC(z))=C(z). The coefficients κn in the Taylor expansion R(z)=n=1κnzn are called free cumulants of the sequence cn. If cn are moments of a probability distribution on R, i.e. cn=Rxnμ(dx), n=0,1,, then R (resp κn) is called the free R-transform (resp. the free cumulants) of μ. If R1(z),R2(z),R(z) are R-transforms of probability

Compositions and monotone convolution

For compactly supported probability distributions μ1,μ2 on R, with the moment generating functionsCμi(z):=Rμi(dx)1xz=n=0Rxnμi(dx) there exists unique compactly supported probability distribution μ1μ2 on R, called monotone convolution of μ1 and μ2, such that its moment generating function satisfiesCμ1μ2(z)=Cμ1(zCμ2(z))Cμ2(z). For details we refer to [14]. In terms of the upshifted moment generating functions Dμi(z):=zCμi(z),Dμ1μ2(z):=zCμ1μ2(z) relation (16) becomesDμ1μ2(z)=Dμ1(Dμ2(z)).

The case r=2: Catalan numbers

For the sake of completeness we briefly describe the case r=2. If aR{0} thencn(a)=au1,u20u1+u2=n1cu1(a)cu2(a),Ca(z)=1+azCa(z)2, which leads to the dilated Catalan numberscn(a)=(2n+1n)an2n+1, with the generating functionCa(z)=114az2az=21+14az. It is well know that the Catalan sequence is positive definite, namely(2n+1n)12n+1=12π04xn4xxdx, n=0,1,. Taking P(w):=ww2 we can now prove this applying Theorem 2.6.

It is also known that μ(1) (and hence μ(a), with a0) is infinitely divisible

The case r=3

In this section we will confine ourselves to r=3. Put a:=a2, b:=a30. The corresponding sequence, polynomial and the generating functions will be denoted cn(a,b), Pa,b(w), Ca,b(z), Da,b(z).

The case r=4

Put a:=(a,b,e)R3, with e0, Pa,b,e(w)=waw2bw3ew4. We have c0(a)=1, c1(a)=a, c2(a)=2a2+b, c3(a)=5a3+5ab+e, c4(a)=14a4+21a2b+3b2+6ae.

Proposition 7.1

If the sequence cn(a,b,e) is positive definite thena6+3a4b+3a2b2+2b32abee20.

Sincea6+3a4b+3a2b2+2b32abee2=(a2+b)3+(a2+b)b2(ab+e)2, inequality (34) implies that a2+b0.

Proof

Inequality (34) is equivalent to c0(a)c2(a)c1(a)20, which is nonnegativity of the second Hankel determinant. 

Symmetric case

In this section we assume that aj=0 whenever j is even, i.e. that the polynomial Pa(w) is odd. Then Da(z) is odd as well, Ca(z) is even and cn(a)=0 whenever n is odd. If in addition the sequence cn(a) is positive definite then the corresponding probability distribution μ(a) is symmetric, i.e. μ(a)(X)=μ(a)(X) for every Borel set XR. The case r=3 was already studied in Subsection 6.6.

Examples

In this section we provide a record of examples of sequences cn(a) for which we can verify positive definiteness. Some of them appear in OEIS with an additional 0 or 1 term at the beginning. We refer also to Table 5 in [9]. To the best of our knowledge, positive definiteness of the sequences listed in subsections 9.1 (except A048779 and A001700), 9.3 (except A006013), 9.4 (except A006632) and 9.5 is the result of this paper.

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W.M. is supported by the Polish National Science Center grant No. 2016/21/B/ST1/00628.

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