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A332277
Number of widely totally normal integer partitions of n.
14
1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580
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.
Sequence in context: A093354 A374032 A317245 * A109701 A124751 A103374
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 12 2020
EXTENSIONS
a(61)-a(66) from Jinyuan Wang, Jun 26 2020
STATUS
approved