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 A320387 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part. 5
 1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 3, 5, 7, 4, 7, 8, 6, 8, 11, 7, 9, 13, 9, 11, 16, 12, 15, 18, 13, 17, 20, 17, 21, 24, 19, 24, 30, 22, 28, 34, 26, 34, 38, 30, 37, 43, 37, 42, 48, 41, 50, 58, 48, 55, 64, 53, 64, 71, 59, 73, 81, 69, 79, 89, 79, 90, 101, 87, 100, 111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order. LINKS Seiichi Manyama, Table of n, a(n) for n = 0..300 EXAMPLE There are a(29) = 15 such partitions of 29: 01: [29] 02: [10, 19] 03: [11, 18] 04: [12, 17] 05: [13, 16] 06: [14, 15] 07: [5, 10, 14] 08: [6, 10, 13] 09: [6, 11, 12] 10: [7, 10, 12] 11: [8, 10, 11] 12: [3, 6, 9, 11] 13: [5, 7, 8, 9] 14: [2, 4, 6, 8, 9] 15: [3, 5, 6, 7, 8] There are a(30) = 18 such partitions of 30: 01: [30] 02: [10, 20] 03: [11, 19] 04: [12, 18] 05: [13, 17] 06: [14, 16] 07: [5, 10, 15] 08: [6, 10, 14] 09: [6, 11, 13] 10: [7, 10, 13] 11: [7, 11, 12] 12: [8, 10, 12] 13: [3, 6, 9, 12] 14: [9, 10, 11] 15: [4, 7, 9, 10] 16: [2, 4, 6, 8, 10] 17: [6, 7, 8, 9] 18: [4, 5, 6, 7, 8] PROG (Ruby) def partition(n, min, max)   return [[]] if n == 0   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).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 A320387(n)   (0..n).map{|i| f(i)} end p A320387(50) CROSSREFS Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320388. Sequence in context: A111725 A302257 A324748 * A304707 A112218 A172366 Adjacent sequences:  A320384 A320385 A320386 * A320388 A320389 A320390 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 12 2018 STATUS approved

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Last modified October 1 17:45 EDT 2020. Contains 337444 sequences. (Running on oeis4.)