1 2 Journal of Integer Sequences, Vol. 8 (2005), 3 Article 05.4.4 47 6 23 11 A Catalan Transform and Related Transformations on Integer Sequences Paul Barry School of Science Waterford Institute of Technology Ireland [email protected] Abstract We introduce and study an invertible transformation on integer sequences related to the Catalan numbers. Transformation pairs are identi¯ed among classical sequences. A closely related transformation which we call the generalized Ballot transform is also studied, along with associated transformations. Results concerning the Fibonacci, Jacobsthal and Pell numbers are derived. Finally, we derive results about combined transformations. 1 Introduction In this note, we report on a transformation of integer sequences that might reasonably be called the Catalan transformation. It is easy to describe both by formula (in relation to the general term of a sequence) and in terms of its action on the ordinary generating function of a sequence. It and its inverse can also be described succinctly in terms of the Riordan group. Many classical `core' sequences can be paired through this transformation. It is also linked to several other known transformations, most notably the binomial transformation. Unless otherwise stated, the integer sequences we shall study will be indexed by N, the nonnegative integers. Thus the Catalan numbers, with general term C(n), are described by 1 2n C(n) = n + 1 n µ ¶ with ordinary generating function given by 1 p1 4x c(x) = ¡ ¡ 2x 1 In the following, all sequences an will have o®set 0, that is, they begin a0; a1; a2; : : :. We use n the notation 1 to denote the all 10s sequence 1; 1; 1; : : : with ordinary generating function 1=(1 x) and 0n to denote the sequence 1; 0; 0; 0; : : : with ordinary generating function 1. ¡ n 0 This is A000007. We have 0 = ±n;0 = n as an integer sequence. This notation allows us to regard : : : ( 2)n; ( 1)n; 0n; 1n; 2n; : : : as a sequence of successive binomial transforms (see ¡ ¡ ¡ ¢ next section). In order to characterize the e®ect of the so-called Catalan transformation, we shall look at its e®ect on some common sequences, including the Fibonacci and Jacobsthal numbers. The Fibonacci numbers [17] are amongst the most studied of mathematical objects. They are easy to de¯ne, and are known to have a rich set of properties. Closely associated to the Fibonacci numbers are the Jacobsthal numbers [18]. In a sense that will be made exact below, they represent the next element after the Fibonacci numbers in a one-parameter family of linear recurrences. These and many of the integer sequences that will be encountered in this note are to be found in The On-Line Encylopedia of Integer Sequences [9], [10]. Sequences in this database will be referred to by their Annnnnn number. For instance, the Catalan numbers are A000108. The Fibonacci numbers F(n) A000045 are the solutions of the recurrence an = an 1 + an 2; a0 = 0; a1 = 1 (1) ¡ ¡ with F(n) : 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; : : : (2) The Jacobstahl numbers J(n) A001045 are the solutions of the recurrence an = an 1 + 2an 2; a0 = 0; a1 = 1 (3) ¡ ¡ with J(n) : 0; 1; 1; 3; 5; 11; 21; 43; 85; 171; 341; 683; 1365; : : : (4) 2n ( 1)n J(n) = ¡ (5) 3 ¡ 3 When we change the initial conditions to a0 = 1; a1 = 0, we get a sequence which we will denote by J1(n) A078008, given by J1(n) : 1; 0; 2; 2; 6; 10; 22; 42; 86; 170; 342 : : : (6) 2n 2( 1)n J (n) = + ¡ (7) 1 3 3 We see that n 2 = 2J(n) + J1(n) (8) The Jacobsthal numbers are the case k = 2 for the one-parameter family of recurrences an = an 1 + kan 2; a0 = 0; a1 = 1 (9) ¡ ¡ 2 where the Fibonacci numbers correspond to the case k = 1. The Pell numbers Pell(n) A000129 are the solutions of the recurrence an = 2an 1 + an 2; a0 = 0; a1 = 1 (10) ¡ ¡ with Pell(n) : 0; 1; 2; 5; 12; 29; 70; 169; : : : (11) The Pell numbers are the case k = 2 of the one-parameter family of recurrences an = kan 1 + an 2; a0 = 0; a1 = 1 (12) ¡ ¡ where again the Fibonacci numbers correspond to the case k = 1. 2 Transformations and the Riordan Group We shall introduce transformations that operate on integer sequences. An example of such a transformation that is widely used in the study of integer sequences is the so-called binomial transform [15], which associates to the sequence with general term an the sequence with general term bn where n n b = a (13) n k k Xk=0 µ ¶ If we consider the sequence to be the vector (a0; a1; : : :) then we obtain the binomial transform of the sequence by multiplying this (in¯nite) vector with the lower-triangle matrix Bin whose i (i; j)-th element is equal to j : ¡ ¢ 1 0 0 0 0 0 : : : 1 1 0 0 0 0 : : : 0 1 1 2 1 0 0 0 : : : B 1 3 3 1 0 0 : : : C Bin = B C B 1 4 6 4 1 0 : : : C B C B 1 5 10 10 5 1 : : : C B C B . C B . .. C B C @ A Note that we index matrices starting at (0; 0). This transformation is invertible, with n n n k a = ( 1) ¡ b (14) n k ¡ k Xk=0 µ ¶ We note that Bin corresponds to Pascal's triangle. Its row sums are 2n, while its diagonal sums are the Fibonacci numbers F(n + 1). If (x) is the ordinary generating function of the sequence a , then the generating func- A n tion of the transformed sequence bn is (1=(1 x)) (x=(1 x)). The Riordan group is a set of in¯nite lo¡wer-triangularA ¡ matrices, where each matrix is 2 2 de¯ned by a pair of generating functions g(x) = 1+g1x+g2x +: : : and f(x) = f1x+f2x +: : : 3 where f1 = 0 [8]. The associated matrix is the matrix whose k-th column is generated by g(x)f(x)k6 (the ¯rst column being indexed by 0). The matrix corresponding to the pair g; f is denoted by (g; f) or (g; f). The group law is then given by R (g; f) (h; l) = (g(h f); l f) ¤ ± ± 1 The identity for this law is I = (1; x) and the inverse of (g; f) is (g; f)¡ = (1=(g f¹); f¹) ± where f¹ is the compositional inverse of f. If M is the matrix (g; f), and a = an is an integer sequence with ordinary generating function (x), then the sequence Mafhasgordinary generating function g(x) (f(x)). A 1 x A For example, the Binomial matrix Bin is the element ( 1 x ; 1 x ) of the Riordan group, 1 x ¡ ¡ while its inverse is the element ( 1+x ; 1+x ). It can be shown more generally [11] that the matrix n+ak m m+1 b a b with general term m+bk is the element (x =(1 x) ; x ¡ =(1 x) ) of the Riordan group. This result will be used in a later section, along¡with characterizations¡ of terms of the form 2n+ak ¡ ¢ n+bk [11]. As an example, we cite the result that the lower triangular matrix with general 2n 2n 1 1 2x p1 4x 1 2 term = is given by the Riordan array ( ; ¡ ¡ ¡ ) = ( ; xc(x) ) ¡ ¢n k n+k p1 4x 2x p1 4x ¡ 1 p1 4x ¡ ¡ where¡c(x)¢= ¡¡ 2x¡¢ is the generating function of the Catalan numbers. 1 2 1 2 2 A lower-triangular matrix that is related to ( p1 4x ; xc(x) ) is the matrix ( p1 4x ; x c(x) ). This is no longer a proper Riordan array: it is a str¡etched Riordan array, as describ¡ ed in [1]. 1 2 The row sums of this array are then the diagonal sums of ( p1 4x ; xc(x) ), and hence have n=2 2(n k) ¡ b c expression k=0 n¡ . P ¡ ¢ 3 The Catalan transform We initially de¯ne the Catalan transformation by its action on ordinary generating functions. For this, we let (x) be the generating function of a sequence. The Catalan transform of A that sequence is de¯ned to be the sequence whose generating function is (xc(x)). The Catalan transform thus corresponds to the element of the Riordan group givenA by (1; xc(x)). That this transformation is invertible is demonstrated by Proposition 3.1. The inverse of the Catalan transformation is given by (x) (x(1 x)) A ! A ¡ Proof. We prove a more general result. Consider the Riordan matrix (1; x(1 kx)). Let ¡ (g¤; f¹) denote its Riordan inverse. We then have (g¤; f¹)(1; x(1 kx)) = (1; x) ¡ Hence 1 p1 4kx f¹(1 kf¹) = x kf¹2 f¹+ x = 0 f¹ = ¡ ¡ ¡ ) ¡ ) 2k Since g = 1, g¤ = 1=(g f¹) = 1 also, and thus ± 1 1 p1 4kx (1; x(1 kx))¡ = (1; ¡ ¡ ) ¡ 2k 4 Setting k = 1, we obtain 1 1 (1; x(1 x))¡ = (1; xc(x)) (1; xc(x))¡ = (1; x(1 x)) ¡ ) ¡ as required. 1 1 In the sequel,the following identities will be useful: c(x(1 x)) = 1 x and c(x) = 1 xc(x) . In terms of the Riordan group, the Catalan transform¡ and its¡ inverse are thus¡ given by the elements (1; xc(x)) and (1; x(1 x)). The lower-triangular matrix representing the Catalan transformation has the form ¡ 1 0 0 0 0 0 : : : 0 1 0 0 0 0 : : : 0 1 0 1 1 0 0 0 : : : B 0 2 2 1 0 0 : : : C Cat = B C B 0 5 5 3 1 0 : : : C B C B 0 14 14 9 4 1 : : : C B C B .