Bell Number Calculator
Calculate Bell numbers, which count the number of ways to partition a set.
Calculate B_n
B_5
52
Number of ways to partition a set of 5 elements
Bell Sequence
B0=1B1=1B2=2B3=5B4=15B5=52B6=203B7=877B8=4,140B9=21,147B10=115,975B11=678,570
Stirling Decomposition of B_5
B_5 = Σ S(5, k) for k = 0 to 5
S(5,0)=0+S(5,1)=1+S(5,2)=15+S(5,3)=25+S(5,4)=10+S(5,5)=1=52
Bell Triangle
0
1
1
1
2
2
2
3
5
3
5
7
10
15
4
15
20
27
37
52
5
52
67
87
114
151
203
6
203
255
322
409
523
674
877
7
877
1080
1335
1657
2066
2589
3263
4140
Each row starts with the last element of the previous row. B_n is the first element of row n.
Formulas
Stirling Sum
B_n = Σ S(n,k) for k=0 to n
Recurrence
B_{n+1} = Σ C(n,k)×B_k
Dobinski's Formula
B_n = (1/e) × Σ k^n/k!
Exponential GF
Σ B_n×x^n/n! = e^(e^x-1)