Taylor expansions for the generating function of Catalan-like numbers
نویسندگان
چکیده
منابع مشابه
Taylor expansions for the m-Catalan numbers
By a Taylor expansion of a generating function, we mean that the remainder of the expansion is a functional of the generating function itself. In this paper, we consider the Taylor expansion for the generating function Bm(t) of them-Catalan numbers. In order to give combinatorial interpretations of the coefficients of these expansions, we study a new collection of partial Grand Dyck paths, that...
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In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae ...
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In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices. 1. Motivation The Catalan numbers ...
متن کاملAsymptotic Expansions of Central Binomial Coefficients and Catalan Numbers
We give a systematic view of the asymptotic expansion of two well-known sequences, the central binomial coefficients and the Catalan numbers. The main point is explanation of the nature of the best shift in variable n, in order to obtain “nice” asymptotic expansions. We also give a complete asymptotic expansion of partial sums of these sequences.
متن کاملOn Polynomials Related to Derivatives of the Generating Function of Catalan Numbers
In [3] it has been shown that powers of the generating function c(x) of Catalan numbers {QaeNo ft ^ > > > 4 2 , •••}, w h e r e o : = {°, I> •••} (1 4 5 9 a n d A000108 of [8] and references of [3]) can be expressed in terms of a linear combination of 1 and c(x) with coefficients replaced by certain scaled Chebyshev polynomials of the second kind. In this paper, derivatives of c(x) are studied ...
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ژورنال
عنوان ژورنال: Cogent Mathematics
سال: 2016
ISSN: 2331-1835
DOI: 10.1080/23311835.2016.1200305