[seqfan] Seeking proof of a claim in the comments for A000009

[Allan Wechsler]

In the comments for https://oeis.org/A000009, Jon Perry claims:

"Number of partitions of n where if k is the largest part, all parts 1..k
are present."

I wrote some code and verified that this is true, and it's *plausible*, but
does anybody have a proof? It would suffice to put these "gapless"
partitions into correspondence with either partitions into odd parts, or
partitions with distinct parts.