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%I #11 Jan 29 2022 04:20:25
%S 1,1,1,1,2,1,2,2,3,3,3,3,4,6,6,7,8,8,9,11,13,15,18,20,24,25,29,32,39,
%T 42,48,54,63,72,81,89,102,116,132,147,165,187,210,238,264,296,329,371,
%U 414,465,516,580,644,722,803,897,994,1108,1229,1367,1512,1678
%N Number of strict integer partitions of n with no adjacent parts having quotient >= 2.
%C The decapitation of such a partition (delete the greatest part) is term-wise greater than its negated first-differences.
%H Fausto A. C. Cariboni, <a href="/A342097/b342097.txt">Table of n, a(n) for n = 1..400</a>
%e The a(1) = 1 through a(16) = 7 partitions (A..G = 10..16):
%e 1 2 3 4 5 6 7 8 9 A B C D E F G
%e 32 43 53 54 64 65 75 76 86 87 97
%e 432 532 74 543 85 95 96 A6
%e 643 653 654 754
%e 743 753 853
%e 5432 6432 6532
%e 7432
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Thread[Differences[-#]<Rest[#]]&]],{n,30}]
%Y The case of equality (all adjacent parts having quotient 2) is A154402.
%Y The multiplicative version is A342083 or A342084.
%Y The non-strict version allowing quotients of 2 exactly is A342094.
%Y The version allowing quotients of 2 exactly is A342095.
%Y The non-strict version is A342096.
%Y The reciprocal version is A342098.
%Y A000009 counts strict partitions.
%Y A000929 counts partitions with no adjacent parts having quotient < 2.
%Y A003114 counts partitions with adjacent parts differing by more than 1.
%Y A034296 counts partitions with adjacent parts differing by at most 1.
%Y Cf. A027193, A001055, A001227, A003242, A167606, A337135, A342085.
%K nonn
%O 1,5
%A _Gus Wiseman_, Mar 02 2021
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