OFFSET
0,4
COMMENTS
Also the number of reversed strict integer partitions of n with strictly decreasing first quotients.
The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).
LINKS
Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
EXAMPLE
The strict partition (12,10,6,3,1) has first quotients (5/6,3/5,1/2,1/3) so is counted under a(32), even though the differences (-2,-4,-3,-2) are not strictly decreasing.
The a(1) = 1 through a(13) = 12 partitions (A..D = 10..13):
1 2 3 4 5 6 7 8 9 A B C D
21 31 32 42 43 53 54 64 65 75 76
41 51 52 62 63 73 74 84 85
321 61 71 72 82 83 93 94
431 81 91 92 A2 A3
432 541 A1 B1 B2
531 631 542 543 C1
4321 641 642 652
731 651 742
741 751
831 841
5431
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Greater@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 30}]
CROSSREFS
The version for differences instead of quotients is A320388.
The non-strict ordered version is A342494.
The strictly increasing version is A342517.
The weakly decreasing version is A342519.
A045690 counts sets with maximum n with all adjacent elements y < 2x.
A167865 counts strict chains of divisors > 1 summing to n.
A342098 counts (strict) partitions with all adjacent parts x > 2y.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 20 2021
STATUS
approved