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Why am I finding the Catalan numbers in these "Snowball Numbers"?

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I've been having fun trying to find new number systems that aren't in the OEIS.

One such number system, is the "Snowball Numbers", which I will define below.

Apologies if these have been explored before, I could not find them.

While playing with these numbers, I found the Catalan numbers (?!) hidden inside them, and I have no idea why. I was wondering if MathStackExchange might be of help.

Definition

The rules fit in three lines:

Let's denote $\langle d_k \cdots d_1 \rangle$ as a snowball number. For example, ⟨321⟩.

The ones (rightmost) digit is base 2.

Every other digit's base is (the digit to its right) + 2.

A digit's contribution is its value times the product of all the bases to its right.

So the bases aren't fixed, they grow (or shrink) depending on the digits themselves, like a snowball rolling downhill. I write them in angle brackets to keep them apart from ordinary numbers.

Example 1:

⟨321⟩ = 23. Working right to left:

The 1 is base 2 (rule 1) and contributes 1. The 2 is base 3 (its right neighbor is 1, and 1 + 2 = 3). It contributes 2 × (2) = 4, its value times the one base to its right. The 3 is base 4 (2 + 2). It contributes 3 × (3 × 2) = 18 :: its value times the product of both bases to its right.

Total: 1 + 4 + 18 = 23.

Example 2:

⟨7654321⟩ = 40319. The bases come out to 2, 3, 4, 5, 6, 7, 8, so the contributions are

1 + 2·(2) + 3·(3·2) + 4·(4·3·2) + 5·(5·4·3·2) + 6·(6·5·4·3·2) + 7·(7·6·5·4·3·2)

which is 1·1! + 2·2! + ... + 7·7! = 8! − 1 = 40319. The "count down by one" pattern maxes out every base :: it's the snowball equivalent of 999999.

Non-example 3:

⟨20⟩ is not a snowball. Here's the hidden constraint: a digit can be at most one bigger than its right neighbor, or it overflows its base (the 2 would need base ≥ 3, but a right neighbor of 0 only grants base 2). So 4 is actually ⟨100⟩.

⟨1000000⟩ = 64. Over runs of zeros every base collapses to 2, so the system quietly imitates binary until a bigger digit shows up.

1-100 in Snowball

The natural numbers $1-100$ and their snowball representations, reading down each column pair.<br>$$<br>\begin{array}{llcllcllcll}<br>\hline<br>n & \text{snowball} & & n & \text{snowball} & & n & \text{snowball} & & n & \text{snowball} \\<br>\hline<br>1 & \langle 1 \rangle & & 21 & \langle 1011 \rangle & & 41 & \langle 1221 \rangle & & 81 & \langle 11111 \rangle \\<br>2 & \langle 10 \rangle & & 22 & \langle 1210 \rangle & & 42 & \langle 10110 \rangle & & 82 & \langle 12210 \rangle \\<br>3 & \langle 11 \rangle & & 23 & \langle 321 \rangle & & 43 & \langle 10101 \rangle & & 83 & \langle 10121 \rangle \\<br>4 & \langle 100 \rangle & & 24 & \langle 11000 \rangle & & 44 & \langle 12100 \rangle & & 84 & \langle 101100 \rangle \\<br>5 & \langle 21 \rangle & & 25 & \langle 10001 \rangle & & 45 & \langle 2111 \rangle & & 85 & \langle 101001 \rangle \\<br>6 & \langle 110 \rangle & & 26 & \langle 10010 \rangle & & 46 & \langle 3210 \rangle & & 86 & \langle 101010 \rangle \\<br>7 & \langle 101 \rangle & & 27 & \langle 1111 \rangle & & 47 & \langle 1321 \rangle & & 87 & \langle 10211 \rangle \\<br>8 & \langle 1000 \rangle & & 28 & \langle 10100 \rangle & & 48 & \langle 110000 \rangle & & 88 & \langle 121000 \rangle \\<br>9 & \langle 111 \rangle & & 29 & \langle 1021 \rangle & & 49 & \langle 100001 \rangle & & 89 & \langle 3221 \rangle \\<br>10 & \langle 210 \rangle & & 30 & \langle 2110 \rangle & & 50 & \langle 100010 \rangle & & 90 & \langle 21110 \rangle \\<br>11 & \langle 121 \rangle & & 31 & \langle 2101 \rangle & & 51 & \langle 2211 \rangle & & 91 & \langle 21101 \rangle \\<br>12 & \langle 1100 \rangle & & 32 & \langle 100000 \rangle & & 52 & \langle 100100 \rangle & & 92 & \langle 32100 \rangle \\<br>13 & \langle 1001 \rangle & & 33 & \langle 1211 \rangle & & 53 & \langle 10021 \rangle & & 93 & \langle 21011 \rangle \\<br>14 & \langle 1010 \rangle & & 34 & \langle 2210 \rangle & & 54 & \langle 11110 \rangle & & 94 & \langle 13210 \rangle \\<br>15 & \langle 211 \rangle & & 35 & \langle 1121 \rangle & & 55 & \langle 11101 \rangle & & 95 & \langle 3321 \rangle \\<br>16 & \langle 10000 \rangle & & 36 & \langle 11100 \rangle & & 56 & \langle 101000 \rangle & & 96 & \langle 1100000 \rangle \\<br>17 & \langle 221 \rangle & & 37 & \langle 11001 \rangle & & 57 & \langle 11011 \rangle & & 97 & \langle 1000001 \rangle \\<br>18 & \langle 1110 \rangle & & 38 & \langle 11010 \rangle & & 58 & \langle 10210 \rangle & & 98 & \langle 1000010 \rangle \\<br>19 & \langle 1101 \rangle & & 39 & \langle 10011 \rangle & & 59 & \langle 2121 \rangle & & 99 & \langle 12111 \rangle \\<br>20 & \langle 2100 \rangle & & 40 & \langle 21000 \rangle & & 60 & \langle...