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 A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing. 12
 1, 1, 2, 2, 3, 4, 4, 5, 7, 6, 8, 10, 10, 11, 14, 13, 16, 19, 18, 20, 25, 23, 27, 31, 30, 34, 39, 37, 42, 48, 47, 50, 59, 56, 63, 70, 68, 74, 83, 82, 89, 97, 97, 104, 116, 113, 123, 133, 133, 142, 155, 153, 166, 178, 178, 189, 204, 204, 218, 232, 235, 247, 265, 265, 283, 299 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order. 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 strictly decreasing. The Heinz numbers of these partitions are given by A325457. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019 LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000 EXAMPLE There are a(10) = 8 such partitions of 10: 01:  02: [1, 9] 03: [2, 8] 04: [3, 7] 05: [4, 6] 06: [5, 5] 07: [1, 4, 5] 08: [2, 4, 4] There are a(11) = 10 such partitions of 11: 01:  02: [1, 10] 03: [2, 9] 04: [3, 8] 05: [4, 7] 06: [5, 6] 07: [1, 4, 6] 08: [1, 5, 5] 09: [2, 4, 5] 10: [3, 4, 4] MATHEMATICA Table[Length[Select[IntegerPartitions[n], Greater@@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 && ary0.uniq == ary0   }   cnt end def A320470(n)   (0..n).map{|i| f(i)} end p A320470(50) CROSSREFS Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457. Sequence in context: A324744 A097920 A029042 * A320382 A259200 A153155 Adjacent sequences:  A320467 A320468 A320469 * A320471 A320472 A320473 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 13 2018 STATUS approved

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Last modified June 23 17:34 EDT 2021. Contains 345402 sequences. (Running on oeis4.)