Turn any integer into double factorial insight today. Built for homework, research, and quick checks. Download results, verify steps, and avoid calculator mistakes always.
| n | n!! | Notes |
|---|---|---|
| 0 | 1 | By convention. |
| 1 | 1 | Odd base case. |
| 6 | 48 | 6×4×2 |
| 7 | 105 | 7×5×3×1 |
| 10 | 3840 | 10×8×6×4×2 |
The double factorial is defined for integers as: n!! = n × (n−2) × (n−4) × … down to 2 (even) or 1 (odd). By convention, 0!! = 1 and (−1)!! = 1.
For efficient logarithms and very large inputs, we also use closed forms: (2k)!! = 2^k k! and (2k−1)!! = (2^k Γ(k+1/2))/√π. These avoid overflow and enable scientific notation.
Double factorial is written n!! and multiplies integers with the same parity as n. Even n ends at 2; odd n ends at 1. This tool also uses 0!! = 1 and (−1)!! = 1, useful in recursive identities. Examples: 6!! = 48 and 7!! = 105, matching the term lists shown after submission.
Even inputs compress into a factorial: (2k)!! = 2^k·k!. Odd inputs connect to Gamma: (2k−1)!! = (2^k·Γ(k+1/2))/√π. Values surge quickly; 10!! = 3840 while 9!! = 945, so neighboring parities differ substantially. This gap matters in approximations for central binomial coefficients and related bounds.
Exact integers become huge, so the calculator reports ln(n!!) and log10(n!!). Digits are estimated as ⌊log10(n!!)⌋+1, which helps you judge size before forcing exact output. Scientific form m×10^e is derived from log10 and stays stable when exact strings are unwieldy. For comparison, 100!! already has far more digits than typical displays can show.
Double factorials appear in series expansions, spherical harmonics, and integrals of trigonometric powers. They also occur in counting matchings and in formulas for moments of the normal distribution. Analysts often use ratios like (2k)!!/(2k−1)!!, which scale about √(πk). In physics, these quantities arise in Gaussian integrals and in normalization constants for orthogonal polynomials.
For moderate sizes, the calculator multiplies terms exactly, stepping by 2 to preserve parity. If GMP exists, it uses big integers; otherwise it uses safe string multiplication. For large n, lgamma‑based log formulas prevent overflow and still give accurate magnitude estimates. Auto mode selects exact output only within a practical digit limit, while still returning logs and scientific notation for every valid input. This keeps results consistent across servers with different extensions.
The Plotly chart shows log10(i!!) over i from 0 to a computed range, split into even and odd traces. A near‑linear rise on the log scale signals multiplicative growth. Export CSV for spreadsheets or a PDF snapshot for reports, and use the digit estimate to decide whether exact output is necessary. When you change n, the plotted range updates, making it simple to compare growth rates.
n! multiplies every integer from n down to 1. n!! multiplies every other integer (same parity), so it uses steps of 2 and stops at 2 or 1.
These conventions make recurrence relations consistent, especially when converting even and odd products into factorial and Gamma forms. They also simplify combinatorial formulas and avoid special-case breaks in derivations.
Use scientific notation when the estimated digit count is large and the exact value becomes hard to display, copy, or store. The magnitude and exponent remain accurate and easy to compare across inputs.
It supports n ≥ −1 under the integer definition used here. Values below −1 are not computed because the double factorial then depends on extensions that require more careful domain handling.
Double factorials grow extremely fast. A log10 scale keeps values within a readable range, reveals growth trends, and makes even and odd sequences comparable on the same axes.
They are computed from lgamma-based closed forms, which are numerically stable for large inputs. The digit estimate is reliable for scale decisions, while the exact value is shown when feasible.
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