Pascal Triangle Binomial Coefficient in C#

using System;
// Pascal Triangle Binomial Coefficient in C#
//
// 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
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 10.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class Program {
    static void alg_require(bool valid) { if (!valid) throw new ArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int binomial(int n, int k) {
        checked {
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require(((n >= 0) && (n <= 30) && (k >= 0) && (k <= n)));
            int[] row = new int[alg_size((k + 1))];
            row[0] = 1;
            int i = 1;
            int j = 0;
            while ((i <= n)) {
                j = Math.Min(i, k);
                while ((j > 0)) {
                    row[j] = (row[j] + row[(j - 1)]);
                    j = (j - 1);
                }
                i = (i + 1);
            }
            return row[k];
        }
    }

    static void Main() {
        int n = 5;
        int k = 2;
        int result = binomial(n, k);
        Console.WriteLine(result);
    }
}