The number series follows the following rule

1st number = 9
2^nd number = 9x2+1 = 19
3^rd number = 19x2+2 = 40
4^th number = 40x2+3 = 83
5^th number = 83x2+4 = 170

The number series follows the following rule

                        1             5            11            19            29            41

Difference:            4             6              8             10            12

The number series follows the following rule

1st number = 1² = 1
2^nd number = 2³ = 8
3^rd number = 3² = 9
4^th number = 4³ = 64

5^th number = 5² = 25

6^th number = 6³ = 216

The number series is the sequential perfect square of prime numbers.

2²=4, 3²=9, 5²=25, 7²=49, 11²=121, 13²=169

The number series follows the following rule

1st number = 0x5+1 = 1
2^nd number = 1x5+1 = 6
3^rd number = 6x5+1 = 31
4^th number = 31x5+1 = 156

5^th number = 156x5+1 = 781

The number series follows the following rule

1st number = 1 
2^nd number = 1+1 = 2
3^rd number = 2+ 2 = 4
4^th number = 4+3 =  7

5^th number = 7+4 =  11

6^th number = 11+5 =16

The number series follows the following rule

1st number = 1²+ 1 = 2
2^nd number = 2²+ 1 = 5
3^rd number = 3²+ 1 = 10
4^th number = 4²+ 1 = 17

5^th number = 5²+ 1 = 26

6^th number = 6²+1 = 37

The number series follows the following rule

1st number = 1²- 1 = 0
2^nd number = 2²- 1 = 3
3^rd number = 3²- 1 = 9
4^th number = 4²- 1 = 15

5^th number = 5²- 1 = 24

6^th number = 6²- 1 = 35

The number series consists of two separate series

      Series-1:                 5            8            11          14

                    Difference:                          3               3               3

Series-2:                     1²                2²           3²            4²             5²  

The number series consists of two separate series

Series-1:                 2             4             6             8

Difference:                    2             2             2

Series-2:                 3             6           12            24           48

Difference:                (..x 2)       (..x 2)       (..x 2)      (..x 2)