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%I #5 Mar 09 2020 18:25:08
%S 1,2,3,5,7,10,15,20,27,35,49,58,81,100,126,160,206,246,316,374,462,
%T 564,696,813,1006,1195,1441,1701,2058,2394,2896,3367,4007,4670,5542,
%U 6368,7540,8702,10199,11734,13760,15734,18384,21008,24441,27893,32380,36841
%N Number of strictly recursively normal integer partitions of n.
%C A sequence is strictly recursively normal if either it empty, its run-lengths are distinct (strict), or its run-lengths cover an initial interval of positive integers (normal) and are themselves a strictly recursively normal sequence.
%e The a(1) = 1 through a(9) = 15 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (21) (31) (32) (42) (43) (53) (54)
%e (211) (41) (51) (52) (62) (63)
%e (221) (321) (61) (71) (72)
%e (311) (411) (322) (332) (81)
%e (331) (422) (432)
%e (421) (431) (441)
%e (511) (521) (522)
%e (3211) (611) (531)
%e (3221) (621)
%e (4211) (711)
%e (3321)
%e (4221)
%e (4311)
%e (5211)
%e (32211)
%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[ptn=={},UnsameQ@@qtn,And[normQ[qtn],recnQ[qtn]]]];
%t Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]
%Y The narrow instead of strict version is A332272.
%Y A wide instead of strict version is A332295(n) - 1 for n > 1.
%Y Cf. A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332277, A332576.
%K nonn
%O 0,2
%A _Gus Wiseman_, Mar 09 2020
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