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A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.

%I A179254 #52 Feb 20 2023 06:14:51
%S A179254 1,1,1,2,2,3,3,5,5,6,8,9,9,13,14,15,19,21,22,28,30,32,39,42,44,54,58,
%T A179254 61,72,77,82,96,102,108,124,133,141,160,171,180,203,218,230,256,273,
%U A179254 289,320,342,361,395,423,447,486,520,548,594,635,669,721,769,811,871,928,978,1044,1114
%N A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.
%C A179254 Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <  p(k) - p(k-1) for all k >= 3.
%H A179254 Alois P. Heinz, <a href="/A179254/b179254.txt">Table of n, a(n) for n = 0..1000</a> (first 225 terms from Joerg Arndt)
%e A179254 There are a(17) = 21 such partitions of 17:
%e A179254 01:  [ 1 2 4 10 ]
%e A179254 02:  [ 1 2 5 9 ]
%e A179254 03:  [ 1 2 14 ]
%e A179254 04:  [ 1 3 13 ]
%e A179254 05:  [ 1 4 12 ]
%e A179254 06:  [ 1 5 11 ]
%e A179254 07:  [ 1 16 ]
%e A179254 08:  [ 2 3 12 ]
%e A179254 09:  [ 2 4 11 ]
%e A179254 10:  [ 2 5 10 ]
%e A179254 11:  [ 2 15 ]
%e A179254 12:  [ 3 4 10 ]
%e A179254 13:  [ 3 5 9 ]
%e A179254 14:  [ 3 14 ]
%e A179254 15:  [ 4 5 8 ]
%e A179254 16:  [ 4 13 ]
%e A179254 17:  [ 5 12 ]
%e A179254 18:  [ 6 11 ]
%e A179254 19:  [ 7 10 ]
%e A179254 20:  [ 8 9 ]
%e A179254 21:  [ 17 ]
%e A179254 - _Joerg Arndt_, Mar 31 2014
%o A179254 (Sage)
%o A179254 def A179254(n):
%o A179254     has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
%o A179254     allowed = lambda x: len(x) < 3 or has_increasing_diffs(x)
%o A179254     return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
%o A179254 # _D. S. McNeil_, Jan 06 2011
%o A179254 (Ruby)
%o A179254 def partition(n, min, max)
%o A179254   return [[]] if n == 0
%o A179254   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
%o A179254 end
%o A179254 def f(n)
%o A179254   return 1 if n == 0
%o A179254   cnt = 0
%o A179254   partition(n, 1, n).each{|ary|
%o A179254     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A179254     cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
%o A179254   }
%o A179254   cnt
%o A179254 end
%o A179254 def A179254(n)
%o A179254   (0..n).map{|i| f(i)}
%o A179254 end
%o A179254 p A179254(50) # _Seiichi Manyama_, Oct 12 2018
%Y A179254 Cf. A007294, A179255 (nondecreasing differences), A179269, A320382, A320385.
%Y A179254 Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences).
%K A179254 nonn
%O A179254 0,4
%A A179254 _Joerg Arndt_, Jan 05 2011