Bernoulli's triangle is an

array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...

partial sums In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

of the

binomial coefficients In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

. 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: :

\sum_^k ,

i.e., the sum of the first ''k'' ''n''th-order binomial coefficients. The first rows of Bernoulli's triangle are: :

\begin
& k & 0 & 1 & 2 & 3 & 4 & 5\\
n & & \\
\hline
0 & & 1 \\
1 & & 1 & 2 \\
2 & & 1 & 3 & 4 \\
3 & & 1 & 4 & 7 & 8 \\
4 & & 1 & 5 & 11 & 15 & 16 \\
5 & & 1 & 6 & 16 & 26 & 31 & 32
\end

Similarly to

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

, 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

B_

denotes the component in row ''n'' and column ''k'', then: :

\begin
B_=&B_+B_ &\mbox&k

As in Pascal's triangle and other similarly constructed triangles, sums of components along diagonal paths in Bernoulli's triangle result in the

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...

s. As the third column of Bernoulli's triangle (''k'' = 2) is a triangular number plus one, it forms the

lazy caterer's sequence The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cut ...

for ''n'' cuts, where ''n'' ≥ 2. The fourth column (''k'' = 3) is the three-dimensional analogue, known as the

cake number In mathematics, the cake number, denoted by ''Cn'', is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly ''n'' planes. The cake number is so-called because one may imagine each partition of the cub ...

s, 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 The number of and for first 6 terms of Moser's circle problem In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with ''n'' sides in such a way as to ''maximise'' the number of areas created by the edges an ...

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

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s, for ''n'' ≥ ''k''. It also gives the number of

compositions Composition or Compositions may refer to: Arts and literature * Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

(ordered partitions) of ''n'' + 1 into ''k'' + 1 or fewer parts.

References

{{Reflist

External links

* The sequence of numbers formed by Bernoulli's triangle on the

On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...

: https://oeis.org/A008949. Factorial and binomial topics Triangles of numbers