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Number of compositions of n with strictly increasing differences.
12

%I #15 Aug 28 2019 10:27:41

%S 1,1,2,3,6,8,11,18,24,30,45,57,71,96,120,148,192,235,286,354,431,518,

%T 628,752,893,1063,1262,1482,1744,2046,2386,2775,3231,3733,4305,4977,

%U 5715,6536,7507,8559,9735,11112,12608,14252,16177,18265,20553,23204,26090,29223

%N Number of compositions of n with strictly increasing differences.

%C A composition of n is a finite sequence of positive integers summing to n.

%C 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 Andrew Howroyd, <a href="/A325547/b325547.txt">Table of n, a(n) for n = 0..1000</a>

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The a(1) = 1 through a(6) = 11 compositions:

%e (1) (2) (3) (4) (5) (6)

%e (11) (12) (13) (14) (15)

%e (21) (22) (23) (24)

%e (31) (32) (33)

%e (112) (41) (42)

%e (211) (113) (51)

%e (212) (114)

%e (311) (213)

%e (312)

%e (411)

%e (2112)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Differences[#]&]],{n,0,15}]

%o (PARI) \\ Row sums of R(n) give A179269 (breakdown by width)

%o 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 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 Cf. A000079, A000740, A008965, A034297, A070211, A175342, A179269, A179254, A240027, A325545, A325546, A325548, A325552, A325557.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 10 2019

%E a(26)-a(42) from _Lars Blomberg_, May 30 2019

%E Terms a(43) and beyond from _Andrew Howroyd_, Aug 27 2019