%I
%S 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,
%T 13,17,20,17,21,24,19,24,30,22,28,34,26,34,38,30,37,43,37,42,48,41,50,
%U 58,48,55,64,53,64,71,59,73,81,69,79,89,79,90,101,87,100,111
%N Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.
%C Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
%H Fausto A. C. Cariboni, <a href="/A320387/b320387.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..300 from Seiichi Manyama)
%e There are a(29) = 15 such partitions of 29:
%e 01: [29]
%e 02: [10, 19]
%e 03: [11, 18]
%e 04: [12, 17]
%e 05: [13, 16]
%e 06: [14, 15]
%e 07: [5, 10, 14]
%e 08: [6, 10, 13]
%e 09: [6, 11, 12]
%e 10: [7, 10, 12]
%e 11: [8, 10, 11]
%e 12: [3, 6, 9, 11]
%e 13: [5, 7, 8, 9]
%e 14: [2, 4, 6, 8, 9]
%e 15: [3, 5, 6, 7, 8]
%e There are a(30) = 18 such partitions of 30:
%e 01: [30]
%e 02: [10, 20]
%e 03: [11, 19]
%e 04: [12, 18]
%e 05: [13, 17]
%e 06: [14, 16]
%e 07: [5, 10, 15]
%e 08: [6, 10, 14]
%e 09: [6, 11, 13]
%e 10: [7, 10, 13]
%e 11: [7, 11, 12]
%e 12: [8, 10, 12]
%e 13: [3, 6, 9, 12]
%e 14: [9, 10, 11]
%e 15: [4, 7, 9, 10]
%e 16: [2, 4, 6, 8, 10]
%e 17: [6, 7, 8, 9]
%e 18: [4, 5, 6, 7, 8]
%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  1).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 ary << 0
%o ary0 = (1..ary.size  1).map{i ary[i  1]  ary[i]}
%o cnt += 1 if ary0.sort == ary0
%o }
%o cnt
%o end
%o def A320387(n)
%o (0..n).map{i f(i)}
%o end
%o p A320387(50)
%Y Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320388.
%K nonn
%O 0,4
%A _Seiichi Manyama_, Oct 12 2018
