%I #22 Jan 06 2021 10:39:50

%S 1,1,2,3,5,6,10,12,16,21,27,32,43,50,60,75,90,103,128,146,170,203,234,

%T 264,315,355,402,467,530,589,684,764,851,969,1083,1195,1360,1504,1659,

%U 1863,2063,2258,2531,2779,3039,3379,3709,4032,4474,4880,5304,5846,6373,6891,7578,8227,8894,9727,10550,11357,12405,13404,14419

%N Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

%C Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - _Gus Wiseman_, May 03 2019

%H Fausto A. C. Cariboni, <a href="/A240026/b240026.txt">Table of n, a(n) for n = 0..500</a> (terms 0..203 from Joerg Arndt)

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e There are a(10) = 27 such partitions of 10:

%e 01: [ 1 1 1 1 1 1 1 1 1 1 ]

%e 02: [ 1 1 1 1 1 1 1 1 2 ]

%e 03: [ 1 1 1 1 1 1 1 3 ]

%e 04: [ 1 1 1 1 1 1 4 ]

%e 05: [ 1 1 1 1 1 2 3 ]

%e 06: [ 1 1 1 1 1 5 ]

%e 07: [ 1 1 1 1 2 4 ]

%e 08: [ 1 1 1 1 6 ]

%e 09: [ 1 1 1 2 5 ]

%e 10: [ 1 1 1 7 ]

%e 11: [ 1 1 2 6 ]

%e 12: [ 1 1 3 5 ]

%e 13: [ 1 1 8 ]

%e 14: [ 1 2 3 4 ]

%e 15: [ 1 2 7 ]

%e 16: [ 1 3 6 ]

%e 17: [ 1 9 ]

%e 18: [ 2 2 2 2 2 ]

%e 19: [ 2 2 2 4 ]

%e 20: [ 2 2 6 ]

%e 21: [ 2 3 5 ]

%e 22: [ 2 8 ]

%e 23: [ 3 3 4 ]

%e 24: [ 3 7 ]

%e 25: [ 4 6 ]

%e 26: [ 5 5 ]

%e 27: [ 10 ]

%t Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[#]]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)

%o (Ruby)

%o def partition(n, min, max)

%o return [[]] if n == 0

%o [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

%o end

%o def f(n)

%o return 1 if n == 0

%o cnt = 0

%o partition(n, 1, n).each{|ary|

%o ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}

%o cnt += 1 if ary0.sort == ary0.reverse

%o }

%o cnt

%o end

%o def A240026(n)

%o (0..n).map{|i| f(i)}

%o end

%o p A240026(50) # _Seiichi Manyama_, Oct 13 2018

%Y Cf. A240027 (strictly increasing differences).

%Y Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).

%Y Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.

%K nonn

%O 0,3

%A _Joerg Arndt_, Mar 31 2014