%I #12 Jun 26 2020 06:19:19

%S 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,

%T 32,19,25,33,30,28,44,33,43,57,51,60,83,70,83,103,96,97,125,117,134,

%U 157,157,171,226,215,238,278,302,312,359,357,396,450,444,477,580

%N Number of widely totally normal integer partitions of n.

%C 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.

%C Also the number of widely totally normal reversed integer partitions of n.

%e The a(n) partitions for n = 1, 4, 10, 11, 16, 18:

%e 1 211 4321 33221 443221 543321

%e 1111 33211 322211 4432111 4333221

%e 322111 332111 1111111111111111 4432221

%e 1111111111 11111111111 4433211

%e 43322211

%e 44322111

%e 111111111111111111

%t recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];

%t Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

%Y Normal partitions are A000009.

%Y Taking multiplicities instead of run-lengths gives A317245.

%Y Constantly recursively normal partitions are A332272.

%Y The Heinz numbers of these partitions are A332276.

%Y The case of all compositions (not just partitions) is A332279.

%Y The co-strong version is A332278.

%Y The recursive version is A332295.

%Y The narrow version is a(n) + 1 for n > 1.

%Y Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.

%K nonn

%O 0,4

%A _Gus Wiseman_, Feb 12 2020

%E a(61)-a(66) from _Jinyuan Wang_, Jun 26 2020