Pascal’s Triangle
Summary: A triangular table of binomial coefficients in which each row gives the coefficients of ; rows sum to powers of 2 and are symmetric.
Sources: chapter-1.2.10
Last updated: 2026-04-29
Definition
Pascal’s triangle lists the coefficients of the binomial expansion for . Row contains entries. Euler presents the table up to in ch1.2.10-higher-powers-compound. (source: chapter-1.2.10)
The Table
Entry (zero-indexed) of row is the binomial coefficient .
Properties
Symmetry: Each row is a palindrome — . Euler observes that coefficients “increase from the beginning to the middle, and then decrease in the same order.” (source: chapter-1.2.10)
Row sum equals : Setting in shows the sum of row is . Euler verifies this for rows 1–7. (source: chapter-1.2.10)
Pascal’s rule (implicit): Each interior entry equals the sum of the two entries directly above it. This is equivalent to the identity , which underlies the recursive construction of the table.
How to Compute Any Row
Euler’s direct method: for the th row, write the descending fractions . Start with coefficient 1; each subsequent coefficient is the previous one multiplied by the next fraction. This avoids needing all prior rows. (source: chapter-1.2.10)
Connection to Permutations
The permutation interpretation in ch1.2.11-transpositions-and-binomial-proof explains why the entries have these values: counts the distinct arrangements of copies of among letters. (source: chapter-1.2.11)
Negative and Fractional Rows
When is negative or fractional, the same coefficient formula generates an infinite sequence. These are the coefficients of the binomial series treated in ch1.2.12-irrational-powers-infinite-series and ch1.2.13-negative-powers-infinite-series.