Abstract:
In an unbiased approach to biomarker discovery, we applied a highly multiplexed proteomic technology (SOMAscan, SomaLogic, Inc, Boulder, CO) to understand changes in proteins from paired serum samples at enrollment and after 8 weeks of TB treatment from 39 patients with pulmonary TB from Kampala, Uganda enrolled in the Center for Disease Control and Prevention’s Tuberculosis Trials Consortium (TBTC) Study 29. This work represents the first large-scale proteomic analysis employing modified DNA aptamers in a study of active tuberculosis (TB). We identified multiple proteins that exhibit significant expression differences during the intensive phase of TB therapy. There was enrichment for proteins in conserved networks of biological processes and function including antimicrobial defense, tissue healing and remodeling, acute phase response, pattern recognition, protease/anti-proteases, complement and coagulation cascade, apoptosis, immunity and inflammation pathways. Members of cytokine pathways such as interferon-gamma, while present, were not as highly represented as might have been predicted. The top proteins that changed between baseline and 8 weeks of therapy were TSP4, TIMP-2, SEPR, MRC-2, Antithrombin III, SAA, CRP, NPS-PLA2, LEAP-1, and LBP. The novel proteins elucidated in this work may provide new insights for understanding TB disease, its treatment and subsequent healing processes that occur in response to effective therapy.

Abstract:
A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of P_{n,0,p}(x). From this we derive that P_{n,2,2}(x) may be obtain in terms of trigonometric functions, from which we obtain some of its important properties. Some questions about orthogonality are also concerned. Furthermore, it is shown that P_{n,2,2}(x) fulfills the same three terms recurrence as Chebyshev polynomials. Some others recurrences for P_{n,m,p}(x) and its coefficients are also obtained. At the end a formula for coefficients of Chebyshev polynomials of the second kind is derived.

Abstract:
We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is an introduction. In the second section we derive an explicit formula for F. From the expression for the power function we obtain a number theory result. Then we derive a formula which shows that the case of arbitrary m may be reduced to the case m=0. This formula extends Vandermonde convolution. In the second section we describe F by the series of recurrence relations with respect to each of arguments k, n, and P. As a special case of the first recurrence relation we state a binomial identity. As a consequence of the second recurrence relation we obtain relation for coefficients of Chebyshev polynomial of both kind. This means that these polynomials might be defined in pure combinatorial way.

Abstract:
It is shown in this note that non-central Stirling numbers s(n,k,a) of the first kind naturally appear in the expansion of derivatives of the product of a power function and a logarithn function. We first obtain a recurrence relation for these numbers, and then, using Leibnitz rule we obtain an explicit formula for these numbers. We also obtain an explicit formula for s(n,1,a), and then derive several combinatorial identities related to these numbers.

Abstract:
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed formula in which arbitrary term of a homogenous linear recurrence equation is expressed in terms of the initial conditions and the coefficients is proved.

Abstract:
In this paper we consider particular generalized compositions of a natural number with a given number of parts. Its number is a weighted polynomial coefficient. The number of all generalized compositions of a natural number is a weighted $r$-generalized Fibonacci number. A relationship between these two numbers will be derived. We shall thus obtain a generalization of the well-known formula connecting Fibonacci numbers with the binomial coefficients.

Abstract:
We first give a combinatorial interpretation of coefficients of Chebyshev polynomials, which allows us to connect them with compositions of natural numbers. Then we describe a relationship between the number of compositions of a natural number in which a certain number of parts are p-1, and other parts are not less than p with compositions in which all parts are not less than p. Then we find a relationship between principal minors of a type of Hessenberg matrices and compositions of natural numbers.

Abstract:
We consider a particular type of matrices which belong at the same time to the class of Hessenberg and Toeplitz matrices, and whose determinants are equal to the number of a type of compositions of natural numbers. We prove a formula in which the number of weak compositions with a fixed number of zeroes is expressed in terms of the number of compositions without zeroes. Then we find a relationship between weak compositions and coefficients of characteristic polynomials of appropriate matrices. Finally, we prove three explicit formulas for weak compositions of a special kind.

Abstract:
We show that the compositions of positive integers may be interpreted in terms of powers of some power series, over arbitrary commutative ring. As consequences, several closed formulas for the compositions as well as for the generalized compositions with a fixed number of parts are derived. Some results on compositions obtained in some recent papers are consequences of these formulas.

Abstract:
We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of the Catalan triangle.