About: Bernoulli's triangle

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Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is given by: i.e., the sum of the first k nth-order binomial coefficients. The first rows of Bernoulli's triangle are: Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if denotes the component in row n and column k, then:

Attributes	Values
rdfs:label	Bernoulli-Dreieck (de) Bernoulli's triangle (en) Triangle de Bernoulli (fr)
rdfs:comment	Le triangle de Bernoulli est une présentation en tableau triangulaire des sommes partielles des lignes du triangle de Pascal. Il est répertorié comme suite de l'OEIS. (fr) Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is given by: i.e., the sum of the first k nth-order binomial coefficients. The first rows of Bernoulli's triangle are: Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if denotes the component in row n and column k, then: (en)
foaf:depiction
dcterms:subject	Triangles of numbers Factorial and binomial topics
Wikipage page ID	52245438 (xsd:integer)
Wikipage revision ID	1122734050 (xsd:integer)
Link from a Wikipage to another Wikipage	Triangles of numbers Binomial coefficients Factorial and binomial topics Array Cake number Composition (combinatorics) Space (mathematics) Dividing a circle into areas Fibonacci number Pascal's triangle Hyperplane Lazy caterer's sequence On-Line Encyclopedia of Integer Sequences Partial sums
Link from a Wikipage to an external page	https://oeis.org/A008949.
sameAs	Bernoulli's triangle Bernoulli's triangle Bernoulli's triangle Bernoulli's triangle Bernoulli's triangle
dbp:wikiPageUsesTemplate	dbt:Reflist dbt:Short_description
thumbnail	wiki-commons:Special:FilePath/Bernoulli_triangle_derivation.svg?width=300
has abstract	Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0 and n, the component in row n and column k is given by: i.e., the sum of the first k nth-order binomial coefficients. The first rows of Bernoulli's triangle are: Similarly to Pascal's triangle, each component of Bernoulli's triangle is the sum of two components of the previous row, except for the last number of each row, which is double the last number of the previous row. For example, if denotes the component in row n and column k, then: As in Pascal's triangle and other similarly constructed triangles, sums of components along diagonal paths in Bernoulli's triangle result in the Fibonacci numbers. As the third column of Bernoulli's triangle (k = 2) is a triangular number plus one, it forms the lazy caterer's sequence for n cuts, where n ≥ 2. The fourth column (k = 3) is the three-dimensional analogue, known as the cake numbers, for n cuts, where n ≥ 3. The fifth column (k = 4) gives the maximum number of regions in the problem of dividing a circle into areas for n + 1 points, where n ≥ 4. In general, the (k + 1)th column gives the maximum number of regions in k-dimensional space formed by n − 1 (k − 1)-dimensional hyperplanes, for n ≥ k. It also gives the number of compositions (ordered partitions) of n + 1 into k + 1 or fewer parts. (en) Le triangle de Bernoulli est une présentation en tableau triangulaire des sommes partielles des lignes du triangle de Pascal. Il est répertorié comme suite de l'OEIS. (fr)
prov:wasDerivedFrom	wikipedia-en:Bernoulli's_triangle?oldid=1122734050&ns=0
page length (characters) of wiki page	4067 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Bernoulli's_triangle
is Link from a Wikipage to another Wikipage of	Bernoulli family Cake number Dividing a circle into areas Pascal's triangle Bernoulli Lazy caterer's sequence List of things named after members of the Bernoulli family Bernoulli triangle
is Wikipage redirect of	Bernoulli triangle
is Wikipage disambiguates of	Bernoulli
is foaf:primaryTopic of	wikipedia-en:Bernoulli's_triangle

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