Make me a metasequence

Wolfram Language (Mathematica), 34 bytes

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

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

Haskell, 34 bytes

(iterate(init.scanl(+)1)[1..20]!!)

Uses 0-indexed inputs (f 4 returns tier 5.)

Haskell, 36 bytes

f 1=[1..20]
f n=init$scanl(+)1$f$n-1

Try it online! Uses 1-indexed inputs (f 5 returns tier 5.)

Explanation

scanl (+) 1 is a function that takes partial sums of a list, starting from (and prepending) 1.

For example: scanl (+) 1 [20,300,4000] equals [1,21,321,4321].

It turns out that tier \$n\$ is just this function applied \$ (n-1) \$ times to the list \$[1,2,3,\dots]\$.

(Or equivalently: \$n\$ times to a list of all ones.)

We use either init or [1..20-n] to account for the list getting longer by \$1\$ every application.

Jelly, 8 7 bytes

20ḶcþŻS

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   cþ       Table of binom(x,y) where:
20Ḷ           x = [0..19]
     Ż        y = [0..n]    e.g.  n=3 → [[1, 1, 1, 1, 1, 1,  …]
                                         [0, 1, 2, 3, 4, 5,  …]
                                         [0, 0, 1, 3, 6, 10, …]
                                         [0, 0, 0, 1, 4, 10, …]]

      S     Columnwise sum.           →  [1, 2, 4, 8, 15, 26, …]

This uses @alephalpha’s insight that $$\text{meta-sequence}_n(i) = \sum_{k=0}^n \binom ik.$$