Make me a metasequence Background
For this challenge, a 'metasequence' will be defined as a sequence of numbers where not only the numbers themselves will increase, but also the increment, and the increment will increase by an increasing value, etc.
For instance, the tier 3 metasequence would start as:
1 2 4 8 15 26 42 64 93 130 176
because:
    1 2 3  4  5  6  7  8   9       &gt;-|
      ↓+↑ = 7                        | Increases by the amount above each time
  1 2 4 7  11 16 22 29 37  46  &gt;-| &lt;-|
                                 | Increases by the amount above each time
1 2 4 8 15 26 42 64 93 130 176 &lt;-|

Challenge
Given a positive integer, output the first twenty terms of the metasequence of that tier.
Test cases
Input: 3 Output: [ 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160 ]
Input: 1 Output: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 ]
Input: 5 Output: [ 1, 2, 4, 8, 16, 32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664 ]
Input: 13 Output: [ 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16383, 32752, 65399, 130238, 258096, 507624 ]
As you may realise, the first \$t+1\$ items of each sequence of tier \$t\$ are the first \$t+1\$ powers of 2...
Rules

Standard loopholes apply
This is code-golf, so shortest answer in bytes wins
Wolfram Language (Mathematica), 34 bytes

0~Range~19~Binomial~i~Sum~{i,0,#}&amp;


Try it online!

The tier \$n\$ metasequence is the sum of the first \$n+1\$ elements of each row of the Pascal triangle.