So far I could identify the sequences up to the level 3 (from 0 to 3):
- Level 0: {1, 1, 1, 1, 1, ...} it's a boring sequence of 1s, nothing new (it has even an OEIS sequence: A000012)
- Level 1: {1, 2, 3, 4, 5, ...} the natural numbers (n ÷ 1), also nothing new (OEIS A000027)
- Level 2: {1, 5, 15, 34, 65, ...} a third-level polynomial, n × (n² + 1) ÷ 2 (A006003)
- Level 3: {1, 20, 210, 1309, 5720, ...} n × (n² + 1)((n² + 2)² - n²) ÷ 16 (A094311)
- Level 4: {1, 230, 26565, 1159774, 25170860, ...} doesn't exist on OEIS.
- Level 5 and above: also don't exist on OEIS.