Pascal's triangle is a number triangle with binomial coefficients arranged in staggered rows such that
 |
(1)
|
where 
and 
are integers with 
and 
, and 
is a binomial coefficient.
The triangle was studied by B. Pascal, in whose posthumous work it appeared
in 1665 (Pascal 1665). However, it had been previously investigated by many other
mathematicians, including Italian algebraist Niccolò Tartaglia, who published
the first six rows of the triangle in 1556. It was also described centuries earlier
by Chinese mathematician Yang Hui and the Persian astronomer-poet Omar Khayyám.
As a result, it is known as the Yang Hui triangle in China, the Khayyam triangle
in Persia, and Tartaglia's triangle in Italy.
Starting with 
,
the number triangle is
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(2)
|
(OEIS A007318). Pascal's formula shows that each subsequent row is obtained by adding the two entries
diagonally above, with an entry outside the triangle treated as 0. For 
,
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(3)
|
Replacing the binomial coefficients in Pascal's triangle by multinomial coefficients in three indices gives Pascal's tetrahedron, a three-dimensional analog in which each interior entry is the sum of the three adjacent entries above it.
The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Pascal's triangle.
For each row indexed by 
, the first entry after the initial 1 divides every interior
entry in that row iff 
is a prime.
The sums 
of the number of odd entries
in the first 
rows of Pascal's triangle for 
, 1, ... are 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45,
49, ... (OEIS A006046). For 
, it is then true that
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(4)
|
(Harborth 1977, Le Lionnais 1983), with equality for 
a power of 2, and the exponent
of 
given by the constant
 |
(5)
|
(OEIS A020857). The infimum limit 
of the normalized cumulative counts 
as 
(OEIS A077464)
is known as the Stolarsky-Harborth constant.
Pascal's triangle contains the figurate numbers along its diagonals, as can be seen from the identity, valid for integers 
,
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(6)
|
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(7)
|
In addition, the sum of the elements of row 
is
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(8)
|
so for a positive integer 
, the sum of the first 
rows (i.e., rows 0 to 
) is the Mersenne number
 |
(9)
|
The shallow diagonals of Pascal's triangle sum to Fibonacci numbers, i.e.,
 |  |  |
(10)
|
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(11)
|
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(12)
|
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(13)
|
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(14)
|
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(15)
|
and, in general,
 |
(16)
|
where 
is a nonnegative integer, 
is a Fibonacci number,
and 
is the floor function applied to 
.
The numbers of times that the numbers 2, 3, 4, ... occur in Pascal's triangle are given by 1, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 4, ... (OEIS A003016; Singmaster 1971; Ogilvy 1972, p. 96; Comtet 1974, p. 93). Similarly, the numbers of rows in which the numbers 2, 3, 4, ... occur are 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, ... (OEIS A059233).
By row 210, the numbers
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(17)
|
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(18)
|
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(19)
|
have appeared six times, more than any other number (excluding 1). By row 1540,
 |
(20)
|
has now occurred six times, by row 3003,
 |
(21)
|
has now occurred 8 times, and by row 7140, 7140 has appeared six times as well. In fact, the numbers that occur five or more times in Pascal's triangle are 1, 120,
210, 1540, 3003, 7140, 11628, 24310, ... (OEIS A003015),
with no others up to 
(de Weger 1997).
There are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to
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(22)
|
given by
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(23)
|
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(24)
|
where 
is a positive integer and 
is the 
th Fibonacci number (Singmaster
1975). The first few such values of 
for 
, 2, ... are 1, 3003, 61218182743304701891431482520, ...
(OEIS A090162).
There is an unexpected connection between Pascal's triangle and the Delannoy numbers via Cholesky decomposition (G. Helms, pers. comm., Aug. 29, 2005). Pascal's triangle and the rascal triangle share their first four rows and the 1s along their edges. The latter triangle provides a connection to the cake cutting problem and hence to the cake numbers.
Pascal's triangle modulo 2 turns out to be equivalent to the Sierpiński sieve (Wolfram 1984, Crandall and Pomerance 2001; Borwein and Bailey 2003, pp. 46-47). Guy (1990) gives several other unexpected properties of Pascal's triangle.
For a nonnegative integer 
, let the set of distinct entries in
row 
be
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(25)
|
For a positive integer 
, let 
be the minimum least
common multiple of 
distinct entries selected from a single row, with the row
also free. Huber (2026) proved
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(26)
|
for some absolute constant 
. Here 
denotes asymptotic notation.
In particular,
 |
(27)
|
Every optimal row index is 
plus an error of the same order. Exact computation shows
that 
is the first case in which no optimal selection consists of the first 
distinct row entries. For a selection with indices 
, its prefix defect is 
, the number of missing indices
up to 
.
The first 
for which every optimal selection has prefix defect greater than 1 is 
. Huber (2026) reports using ChatGPT for exploratory mathematical
discussion and code drafting, followed by review and verification of the mathematical
content and references.
