Statements (62)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
integer sequence
|
| gptkbp:appearsIn |
gptkb:OEIS
|
| gptkbp:author |
gptkb:N._J._A._Sloane
|
| gptkbp:citation |
gptkb:A001047
gptkb:A000594 gptkb:A000292 gptkb:A000389 gptkb:A000579 gptkb:A001405 gptkb:A007318 A000217 A000332 A000918 A000984 |
| gptkbp:columnSumSequence |
gptkb:A000012
|
| gptkbp:describedBy |
Triangle read by rows: T(n, k) = binomial(n, k), 0 <= k <= n.
|
| gptkbp:first_terms |
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, ...
|
| gptkbp:form |
T(n, k) = n! / (k! * (n-k)!)
|
| gptkbp:hasKeyword |
easy
nice core nonn tabl |
| https://www.w3.org/2000/01/rdf-schema#label |
OEIS A007318
|
| gptkbp:mapleCode |
seq(seq(binomial(n, k), k=0..n), n=0..10)
|
| gptkbp:mathematicaCode |
Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}]
|
| gptkbp:name |
gptkb:Pascal's_triangle
|
| gptkbp:OEIS |
gptkb:A007318
|
| gptkbp:property |
symmetric
Lucas' theorem applies central binomial coefficients in center column diagonals give figurate numbers each entry is sum of two above entries appear in binomial expansions entries are coefficients in (x+y)^n entries are log-concave in each row entries are nonnegative integers entries are unimodal in each row entries count lattice paths entries count subsets of n-element set entries count ways to choose k from n first and last entry in each row is 1 mod 2 gives Sierpinski triangle row sums are powers of 2 |
| gptkbp:pythonCode |
[[math.comb(n, k) for k in range(n+1)] for n in range(11)]
|
| gptkbp:relatedTo |
binomial coefficients
combinatorics |
| gptkbp:rowFormula |
Row n: binomial(n, 0), binomial(n, 1), ..., binomial(n, n)
|
| gptkbp:rowSumSequence |
gptkb:A000079
|
| gptkbp:sequence |
gptkb:train
|
| gptkbp:usedIn |
gptkb:Fibonacci_numbers
gptkb:algebra gptkb:binomial_theorem gptkb:probability_theory gptkb:Catalan_numbers gptkb:Sierpinski_triangle gptkb:Pascal's_rule number theory combinatorics polynomial expansions |
| gptkbp:bfsParent |
gptkb:OEIS_A001349
|
| gptkbp:bfsLayer |
7
|