A multidimensional extension of Pascal's triangle

Counting: Basic Subsets of a Set Pascal's Triangle Binomial Coefficient Identities Counting: Intermediate Finding a Polynomial The Upward-Extended Pascal's Triangle Recurrence Relations and Fibonacci Numbers Counting: Advanced Generating Functions and Making Change Integer Triangles Rook Paths and Queen Paths Discrete Probability Probability Spaces and Distributions Markov Chains Random Tournaments Number Theory Divisibility of Factorials and Binomial Coefficients Covering Systems Partitions of an Integer Information Theory What Is Surprise? A Coin-Tossing Game Shannon's Theorems Games A Little Graph Theory Background The Ramsey Game Tic-Tac-Toe and Animal Games Algorithms Counters Listing Permutations and Combinations Sudoku Solving and Polycube Packing Appendix A: Hints and Solutions to Exercises Appendix B: Notation Bibliography Index

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article will give an extension of Essouabri's result to some cases with certain ways to count the points in Pascal's triangle modulo a prime which are different from the traditional one.

In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize connections with the binomial and multinomial theorems, and finally we present some computational results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Mass spectrometry (MS) is widely used for isotopic studies of metabolism in which detailed information about biochemical processes is obtained from the analysis of isotope incorporation into metabolites. The biological value of such experiments is dependent on the accuracy of the isotopic measurements. Using MS, isotopologue distributions are measured from the quantitative analysis of isotopic clusters. These measurements are prone to various biases, which can occur during the experimental workflow and/or MS analysis. The lack of relevant standards limits investigations of the quality of the measured isotopologue distributions. To meet that need, we developed a complete theoretical and experimental framework for the biological production of metabolites with fully controlled and predictable labeling patterns. This strategy is valid for different isotopes and different types of metabolisms and organisms, and was applied to two model microorganisms, Pichia augusta and Escherichia coli, cultivated on (13)C-labeled methanol and acetate as sole carbon source, respectively. The isotopic composition of the substrates was designed to obtain samples in which the isotopologue distribution of all the metabolites should give the binomial coefficients found in Pascal's triangle. The strategy was validated on a liquid chromatography-tandem mass spectrometry (LC-MS/MS) platform by quantifying the complete isotopologue distributions of different intracellular metabolites, which were in close agreement with predictions. This strategy can be used to evaluate entire experimental workflows (from sampling to data processing) or different analytical platforms in the context of isotope labeling experiments.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work. The College Mathematics Journal: Vol. 30, No. 3, pp. 170-186.

The Pascal triangle is a geometric representation of binomial coefficients in triangular form. We utilize this formalism to deterministically arrange silver nanocylinders of different sizes (30, 60, and 90 nm) on a triangle and numerically study their near-field optical properties. We show that near-field intensities at specific points on this triangle depend on the wavelength and angle of incidence. From the wavelength-dependent studies at various junctions of nanocylinders, we obtain maximum near-field intensity at 350 and 380 nm. By varying the angle of incidence of the TM-polarized plane wave, we find systematic variation in the near-field intensity at different junctions of the geometry. Our study will lead to insights in designing controllable electromagnetic hot spots for chip-based plasmonic devices.

Generalized Pascal triangle for binomial coefficients of words

Introduction Blaise Pascal (1623–1662) was born in Clermont-Ferrand in central France. Even as a teenager his father introduced him to meetings for mathematical discussion in Paris run by Marin Mersenne, who served as a primary conduit for transmitting mathematical ideas widely at that time, before the existence of any research journals. He quickly became involved in the development of projective geometry, the first in a sequence of highly creative mathematical and scientific episodes in his life, punctuated by periods of religious fervor. Around age twenty-one he spent several years developing a mechanical addition and subtraction machine, in part to help his father in tax computations as a local administrator. It was the first of its kind ever to be marketed. Then for several years he was at the center of efforts to understand vacuum, which led to an understanding of barometric pressure. In fact the scientific unit of pressure is named the pascal . He is also known for Pascal's Law on the behavior of fluid pressure. Around 1654 Pascal conducted his studies on the Arithmetical Triangle (“Pascal's Triangle”) and its relationship to probabilities. His correspondence with Pierre de Fermat (1601–1665) in that year marks the beginning of probability theory. Several years later, Pascal refined his ideas on area problems via the method of indivisibles already being developed by others, and solved various problems of areas, volumes, centers of gravity, and lengths of curves.

An attractive approach to design antenna arrays with minimum sidelobe level is based on the binomial expansion, and they are referred to as binomial arrays. In fact, binomial arrays with element spacing equal or less than λ/2 have no minor lobes. However, there are no compact expressions available to obtain the directivity of binomial arrays with no restriction in the element spacing, except when the element spacing is λ/2. This letter proposes a compact expression for the directivity of the binomial array with no restriction in the element spacing and number of elements. The procedure is based on the properties of the Pascal's triangle. Then, the array factor and directivity of the binomial array are calculated by using the properties of the Fourier transform.

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that ‘tweaked’ the sum of the sth powers of the n-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call ‘major Apéry miracles’, proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a ‘minor Apéry miracle’ where an explicit constant, defined as an (extremely slowly converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.

Linear cellular automata, finite automata and Pascal's triangle

A property of superposability is defined here for Treillis Automata as an equivalent of the linearity property of Cellular Automata. We take advantage of the binary local transition function of TAs, which enables us to use an infix notation for it, to study this property in an algebraic framework. We show that a TA is superposable iff its local transition function operates on its set of states as a commutative monoid law and that the general case of commutative monoids is reducible to the case of Abelian groups in which the space-time diagrams obtained are isomorphic to a superposition of figures representing the triangle of binomial coefficients (Pascal's triangle) modulo integers.KeywordsCellular AutomatonInitial ConfigurationNeutral ElementBinomial CoefficientCommutative MonoidThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

(2000). The Pop-up Cuboctahedron. The College Mathematics Journal: Vol. 31, No. 2, pp. 89-92.

We propose to construct some triangles somewhat like the Pascal triangle, but more general. The only essential knowledge is that of the binomial theorem, though for some of the results we assume a knowledge of number theory, up to Fermat’s theorem. At the end, we show how matrix algebra could be used to advantage.

Wideband impedance matching of a purely resistive load by means of a cascade of quarter wavelength long transmission lines is dealt with.The problem is formulated as the minimization of the reflection coefficient modulus inside a fixed frequency range at the variance of the intermediate resistances seen at each line section input.Numerical results are shown and compared with matching techniques based on binomial and Chebyshev coefficient's expansion and on the exponential tapered line.