OFFSET
0,4
COMMENTS
A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
Also the number of widely totally normal reversed integer partitions of n.
EXAMPLE
The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
1 211 4321 33221 443221 543321
1111 33211 322211 4432111 4333221
322111 332111 1111111111111111 4432221
1111111111 11111111111 4433211
43322211
44322111
111111111111111111
MATHEMATICA
recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n], recnQ]], {n, 0, 30}]
CROSSREFS
Normal partitions are A000009.
Taking multiplicities instead of run-lengths gives A317245.
Constantly recursively normal partitions are A332272.
The Heinz numbers of these partitions are A332276.
The case of all compositions (not just partitions) is A332279.
The co-strong version is A332278.
The recursive version is A332295.
The narrow version is a(n) + 1 for n > 1.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2020
EXTENSIONS
a(61)-a(66) from Jinyuan Wang, Jun 26 2020
STATUS
approved