Function binomialBigint

src/compute-engine/library/combinatorics.ts:65–89

* Exact binomial coefficient for bigint n, k. * * - k < 0 → 0 (no combinatorial meaning, regardless of n). * - n ≥ 0 and k > n → 0 (standard convention). * - n < 0 → the standard extension via Pascal's rule analytic continuation: * Binomial(n, k) = (-1)^k · Binomial(k-n-1, k), e.g. * Binom

(
  n: bigint,
  k: bigint,
  deadline?: number
)

  if (!Number.isFinite(n) || n < 0) return Infinity;
  if (n < 3) return 1;
  const lnN = Math.log(n);
  const lnB = n * lnN - n * Math.log(lnN) - n;
  return lnB > 0 ? lnB / Math.LN10 : 1;
}

/**
 * Exact binomial coefficient for bigint n, k.
 *
 * - k < 0 → 0 (no combinatorial meaning, regardless of n).
 * - n ≥ 0 and k > n → 0 (standard convention).
 * - n < 0 → the standard extension via Pascal's rule analytic continuation:
 *   Binomial(n, k) = (-1)^k · Binomial(k-n-1, k), e.g.
 *   Binomial(-2, 3) = (-1)³·Binomial(4, 3) = -4 (matches Mathematica/sympy).
 *
 * Returns `undefined` (stay symbolic) rather than an exact bigint when the
 * result would exceed MAX_EXACT_COMBINATORICS_DIGITS decimal digits — e.g.
 * `Binomial(2e9, 1e9)` has ~6×10⁸ digits, pathological to build.
 */
function binomialBigint(
  n: bigint,
  k: bigint,
  deadline?: number
): bigint | undefined {
  if (k < 0n) return 0n;
  if (n < 0n) {
    const sign = k % 2n === 0n ? 1n : -1n;
    const inner = binomialBigint(k - n - 1n, k, deadline);
    return inner === undefined ? undefined : sign * inner;