That's a really bad question, and I'm not just saying that because I don't know what the answer is. The first three sequences are very ordinary mathematical series, suggesting that the fourth would be the same. But the sequence 1, 5, 7, 15, 20 doesn't appear in OEIS[1], let alone 1, 5, 7, 15, 20, 28. I feel comfortable in announcing that if a sequence doesn't appear in OEIS, it isn't mathematically significant. Likewise, "convolution series" doesn't appear to have a well-defined mathematical interpretation.
What sequence is this and what is the justification for 28 being the next number?
Dude. Relax. I said convolution series and not convolution integral ( the one you've linked to ). The "invented term" as you call it appears in Melzak's textbook ( companion to concrete math ). Finally, I actually "invented" a sequence that appears in OEIS ( https://oeis.org/A160138 ) , so I'm well aware that there are infinite sequences you can manufacture by simply changing the op in a convolution series. I just throw these questions into the mix to have some fun and games, otherwise doing interviews becomes very tedious and boring. Nobody is being judged on whether they can figure out that the number of moves for a legitimate tower of hanoi ie. 1,3,7,15, comes from the 2^n-1 sequence ( there's an entire chapter of D.E.Knuth's "concrete math" devoted to just that very topic, and it was my text in undergrad, so I like to have some fun with it. )
What sequence is this and what is the justification for 28 being the next number?
[1] http://oeis.org/