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dplyr is an R package whose set of functions are designed to enable dataframe (a spreadsheet-like data structure) manipulation in an intuitive, user-friendly way. It is one of the core packages of the popular tidyverse set of packages in the R programming language. [1]
The nth row sums to n(n 2 + 1)/2, the constant of an n × n magic square (sequence A006003 in the OEIS). Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers , triangular numbers with an index that is triangular.
R is a programming language for statistical computing and data visualization. It has been adopted in the fields of data mining, bioinformatics and data analysis. [9] The core R language is augmented by a large number of extension packages, containing reusable code, documentation, and sample data. R software is open-source and free software.
Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube. In this triangle, the sum of the elements of row m is equal to 3 m . Again, to use the elements of row 4 as an example: 1 + 8 + 24 + 32 + 16 = 81 , which is equal to 3 4 = 81 {\displaystyle 3^{4}=81} .
The row space is defined similarly. The row space and the column space of a matrix A are sometimes denoted as C(A T) and C(A) respectively. [2] This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and respectively. [3]
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This template will become obsolete if and when MediaWiki supports automatically adding row numbers via WikiTable syntax. See Phab: T42618 - "jquery.tablesorter: Add support for a 'fixed' column of row numbers." And see Help:Sortable tables, and the sections on row numbering and auto-ranking.
Multivariate generating functions arise in practice when calculating the number of contingency tables of non-negative integers with specified row and column totals. Suppose the table has r rows and c columns; the row sums are t 1 , t 2 ...