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Bell Number Calculator

Work out bell number instantly with clear inputs, formula shown and shareable results.

About Bell Number Calculator

The Bell number Bₙ counts all the ways to partition a set of n labelled items into non-empty blocks, with no limit on the number of blocks. The Bell triangle generates them by cumulative row sums.

Formulas

Bell triangle

start a row with the last entry of the previous row, then add the entry above-left repeatedly

From Stirling numbers

Bₙ = Σₖ S(n, k)

Worked example

With the default values below, Bell Number Calculator returns Bell number Bₙ of 4,140. Change any input above and every figure here is recalculated in your browser.

Inputs used

Number of items naccepts 0–22
8
How many Bell numbers to listaccepts 1–20
8

Results

Bell number Bₙ
4,140
Bₙ₊₁
21,147
First Bell numbers
1, 1, 2, 5, 15, 52, 203, 877
Bₙ₊₁ ÷ Bₙ
5.107971

Step by step

  1. 1Values used
  2. 2Bell triangle
  3. 3From Stirling numbers
  4. 4Bell number Bₙ= 4,140
  5. 5Bₙ₊₁= 21,147
  6. 6First Bell numbers= 1, 1, 2, 5, 15, 52, 203, 877
  7. 7Bₙ₊₁ ÷ Bₙ= 5.107971

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is B₈?

4140. The sequence runs 1, 1, 2, 5, 15, 52, 203, 877, 4140.

Why does it grow so fast?

Faster than exponentially but slower than n!, since each new element can join any existing block or open a new one.

Related Calculators

Bell Number Calculator – Calculate Bell number Bₙ & Bₙ₊₁