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 A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part. 14
 1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169 (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. The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (-3,-2,-1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..300 EXAMPLE There are a(11) = 8 such partitions of 11: 01: [11] 02: [4, 7] 03: [5, 6] 04: [2, 4, 5] 05: [3, 4, 4] 06: [2, 3, 3, 3] 07: [1, 2, 2, 2, 2, 2] 08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] There are a(12) = 11 such partitions of 12: 01: [12] 02: [4, 8] 03: [5, 7] 04: [6, 6] 05: [2, 4, 6] 06: [3, 4, 5] 07: [4, 4, 4] 08: [3, 3, 3, 3] 09: [1, 2, 3, 3, 3] 10: [2, 2, 2, 2, 2, 2] 11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] MATHEMATICA Table[Length[Select[IntegerPartitions[n], GreaterEqual@@Differences[Append[#, 0]]&]], {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|     ary << 0     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}     cnt += 1 if ary0.sort == ary0   }   cnt end def A320509(n)   (0..n).map{|i| f(i)} end p A320509(50) CROSSREFS Cf. A240026, A240027, A320466, A320470, A320510. Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part). Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390. Sequence in context: A200763 A203291 A220053 * A159999 A003977 A003971 Adjacent sequences:  A320506 A320507 A320508 * A320510 A320511 A320512 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 14 2018 STATUS approved

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Last modified May 30 09:15 EDT 2020. Contains 334712 sequences. (Running on oeis4.)