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Number of integer partitions of n with no adjacent parts having quotient >= 2.
37

%I #11 Feb 01 2022 05:12:18

%S 1,2,2,3,3,4,4,6,6,8,9,11,13,17,19,24,29,35,42,51,61,75,90,108,130,

%T 158,189,227,272,325,389,464,553,659,782,929,1102,1306,1545,1824,2153,

%U 2538,2989,3514,4127,4842,5673,6642,7766,9068,10583,12335,14361,16705

%N Number of 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="/A342096/b342096.txt">Table of n, a(n) for n = 1..250</a>

%e The a(1) = 1 through a(10) = 8 partitions:

%e 1 2 3 4 5 6 7 8 9 A

%e 11 111 22 32 33 43 44 54 55

%e 1111 11111 222 322 53 333 64

%e 111111 1111111 332 432 433

%e 2222 3222 532

%e 11111111 111111111 3322

%e 22222

%e 1111111111

%t Table[Length[Select[IntegerPartitions[n],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 version allowing quotients of 2 exactly is A342094.

%Y The strict case allowing quotients of 2 exactly is A342095.

%Y The strict case is A342097.

%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, A342085, A342191.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 02 2021