Pascal Triangle Binomial Coefficient in Python

from bisect import bisect_left

# Pascal Triangle Binomial Coefficient in Python
#
# PURPOSE
# Pascal identity computes n choose k from neighboring coefficients in the previous row. A
# single row is enough when updated right to left.
#
# METHOD
# Initialize row[0] to one. For each new row, update row[j] from row[j] plus row[j-1] in
# descending j order, preserving the old row values needed later.
#
# COMPLEXITY
# O(nk) time and O(k) auxiliary space.
#
# CONTRACT
# n >= 0 and 0 <= k <= n. All coefficients computed must fit the integer type.
# Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
# counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
# Sorted order, correct tree topology, and caller-owned data-structure invariants are
# preconditions; checking them on every query would change query complexity.
# Scalar and shape checks are included. Value-range checks are included where needed to
# prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
# workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
# uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
# partly updated after arithmetic failure. These are single-threaded, in-memory
# implementations.
#
# LANGUAGE DETAILS
# Python 3.9+: automatic lengths, native lists, arbitrary-precision arithmetic, and
# bisect_left for LIS. Invalid scalar/shape inputs raise ValueError. Integer element types
# and sorted/tree invariants remain caller preconditions.
#
# EXAMPLE
# Expected return: 10.
# Run: python example.py
# Tests for this exact source artifact run before publication. The accompanying source
# archive includes unit cases, runtime results, and source hashes.

def alg_require(condition):
    if not condition:
        raise ValueError("invalid algorithm input; see contract")
def alg_size(n):
    alg_require(0 <= n <= 4000000)
    return n

def binomial(n: int, k: int) -> int:
    alg_require(type(n) is int and -2147483648 <= n <= 2147483647)
    alg_require(type(k) is int and -2147483648 <= k <= 2147483647)
    alg_require(n >= 0 and n <= 1000000)
    alg_require(n >= 0 and n <= 30 and (k >= 0) and (k <= n))
    row: list[int] = [0] * alg_size(k + 1)
    row[0] = 1
    i: int = 1
    j: int = 0
    while i <= n:
        j = min(i, k)
        while j > 0:
            row[j] += row[j - 1]
            j -= 1
        i += 1
    return row[k]

n = 5
k = 2
result = binomial(n, k)
print(result)