This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A342513 #12 Feb 16 2025 08:34:01
%S A342513 1,1,2,3,4,5,7,8,9,12,13,15,20,21,24,28,29,33,40,44,49,57,61,65,77,84,
%T A342513 87,99,106,115,132,141,152,167,180,193,212,228,246,274,290,309,338,
%U A342513 357,382,412,439,463,498,536,569,608,648,693,743,790,839,903,949
%N A342513 Number of integer partitions of n with weakly decreasing first quotients.
%C A342513 Also called log-concave-down partitions.
%C A342513 Also the number of reversed integer partitions of n with weakly decreasing first quotients.
%C A342513 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).
%H A342513 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LogarithmicallyConcaveSequence.html">Logarithmically Concave Sequence</a>.
%H A342513 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%H A342513 Gus Wiseman, <a href="/A069916/a069916.txt">Sequences counting and ranking partitions and compositions by their differences and quotients</a>.
%e A342513 The partition (9,7,4,2,1) has first quotients (7/9,4/7,1/2,1/2) so is counted under a(23).
%e A342513 The a(1) = 1 through a(8) = 9 partitions:
%e A342513 (1) (2) (3) (4) (5) (6) (7) (8)
%e A342513 (11) (21) (22) (32) (33) (43) (44)
%e A342513 (111) (31) (41) (42) (52) (53)
%e A342513 (1111) (221) (51) (61) (62)
%e A342513 (11111) (222) (331) (71)
%e A342513 (321) (421) (332)
%e A342513 (111111) (2221) (431)
%e A342513 (1111111) (2222)
%e A342513 (11111111)
%t A342513 Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Divide@@@Reverse/@Partition[#,2,1]&]],{n,0,30}]
%Y A342513 The ordered version is A069916.
%Y A342513 The version for differences instead of quotients is A320466.
%Y A342513 The weakly increasing version is A342497.
%Y A342513 The strictly decreasing version is A342499.
%Y A342513 The strict case is A342519.
%Y A342513 The Heinz numbers of these partitions are A342526.
%Y A342513 A000005 counts constant partitions.
%Y A342513 A000009 counts strict partitions.
%Y A342513 A000041 counts partitions.
%Y A342513 A000929 counts partitions with all adjacent parts x >= 2y.
%Y A342513 A001055 counts factorizations.
%Y A342513 A003238 counts chains of divisors summing to n - 1 (strict: A122651).
%Y A342513 A074206 counts ordered factorizations.
%Y A342513 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342513 A342094 counts partitions with adjacent parts x <= 2y.
%Y A342513 Cf. A000837, A002843, A003242, A175342, A318991, A318992, A325557, A342527, A342528, A342529.
%K A342513 nonn
%O A342513 1,3
%A A342513 _Gus Wiseman_, Mar 17 2021