cp's OEIS Frontend

A325547 Number of compositions of n with strictly increasing differences.

%I A325547 #15 Aug 28 2019 10:27:41
%S A325547 1,1,2,3,6,8,11,18,24,30,45,57,71,96,120,148,192,235,286,354,431,518,
%T A325547 628,752,893,1063,1262,1482,1744,2046,2386,2775,3231,3733,4305,4977,
%U A325547 5715,6536,7507,8559,9735,11112,12608,14252,16177,18265,20553,23204,26090,29223
%N A325547 Number of compositions of n with strictly increasing differences.
%C A325547 A composition of n is a finite sequence of positive integers summing to n.
%C A325547 The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
%H A325547 Andrew Howroyd, <a href="/A325547/b325547.txt">Table of n, a(n) for n = 0..1000</a>
%H A325547 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325547 The a(1) = 1 through a(6) = 11 compositions:
%e A325547   (1)  (2)   (3)   (4)    (5)    (6)
%e A325547        (11)  (12)  (13)   (14)   (15)
%e A325547              (21)  (22)   (23)   (24)
%e A325547                    (31)   (32)   (33)
%e A325547                    (112)  (41)   (42)
%e A325547                    (211)  (113)  (51)
%e A325547                           (212)  (114)
%e A325547                           (311)  (213)
%e A325547                                  (312)
%e A325547                                  (411)
%e A325547                                  (2112)
%t A325547 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Differences[#]&]],{n,0,15}]
%o A325547 (PARI) \\ Row sums of R(n) give A179269 (breakdown by width)
%o A325547 R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, v[i-k*t]))); Mat(L)}
%o A325547 seq(n)={my(M=R(n)); Vec(1 + sum(i=1, n, my(p=sum(w=1, min(#M,n\i), x^(w*i)*sum(j=1, n-i*w, x^j*M[j,w])));  x^i*(1 + x^i)*(1 + p + O(x*x^(n-i)))^2))} \\ _Andrew Howroyd_, Aug 27 2019
%Y A325547 Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.
%K A325547 nonn
%O A325547 0,3
%A A325547 _Gus Wiseman_, May 10 2019
%E A325547 a(26)-a(42) from _Lars Blomberg_, May 30 2019
%E A325547 Terms a(43) and beyond from _Andrew Howroyd_, Aug 27 2019