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 A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing. 34
 1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3. 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 LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..203 from Joerg Arndt) EXAMPLE There are a(10) = 27 such partitions of 10: 01:  [ 1 1 1 1 1 1 1 1 1 1 ] 02:  [ 1 1 1 1 1 1 1 1 2 ] 03:  [ 1 1 1 1 1 1 1 3 ] 04:  [ 1 1 1 1 1 1 4 ] 05:  [ 1 1 1 1 1 2 3 ] 06:  [ 1 1 1 1 1 5 ] 07:  [ 1 1 1 1 2 4 ] 08:  [ 1 1 1 1 6 ] 09:  [ 1 1 1 2 5 ] 10:  [ 1 1 1 7 ] 11:  [ 1 1 2 6 ] 12:  [ 1 1 3 5 ] 13:  [ 1 1 8 ] 14:  [ 1 2 3 4 ] 15:  [ 1 2 7 ] 16:  [ 1 3 6 ] 17:  [ 1 9 ] 18:  [ 2 2 2 2 2 ] 19:  [ 2 2 2 4 ] 20:  [ 2 2 6 ] 21:  [ 2 3 5 ] 22:  [ 2 8 ] 23:  [ 3 3 4 ] 24:  [ 3 7 ] 25:  [ 4 6 ] 26:  [ 5 5 ] 27:  [ 10 ] MATHEMATICA Table[Length[Select[IntegerPartitions[n], OrderedQ[Differences[#]]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *) PROG (Ruby) def partition(n, min, max)   return [[]] if n == 0   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}} end def f(n)   return 1 if n == 0   cnt = 0   partition(n, 1, n).each{|ary|     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}     cnt += 1 if ary0.sort == ary0.reverse   }   cnt end def A240026(n)   (0..n).map{|i| f(i)} end p A240026(50) # Seiichi Manyama, Oct 13 2018 CROSSREFS Cf. A240027 (strictly increasing differences). Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing). Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360. Sequence in context: A337218 A306296 A191173 * A213212 A341124 A008627 Adjacent sequences:  A240023 A240024 A240025 * A240027 A240028 A240029 KEYWORD nonn AUTHOR Joerg Arndt, Mar 31 2014 STATUS approved

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Last modified June 24 17:11 EDT 2021. Contains 345417 sequences. (Running on oeis4.)