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A325547
Number of compositions of n with strictly increasing differences.
12
1, 1, 2, 3, 6, 8, 11, 18, 24, 30, 45, 57, 71, 96, 120, 148, 192, 235, 286, 354, 431, 518, 628, 752, 893, 1063, 1262, 1482, 1744, 2046, 2386, 2775, 3231, 3733, 4305, 4977, 5715, 6536, 7507, 8559, 9735, 11112, 12608, 14252, 16177, 18265, 20553, 23204, 26090, 29223
OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
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).
EXAMPLE
The a(1) = 1 through a(6) = 11 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(31) (32) (33)
(112) (41) (42)
(211) (113) (51)
(212) (114)
(311) (213)
(312)
(411)
(2112)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@Differences[#]&]], {n, 0, 15}]
PROG
(PARI) \\ Row sums of R(n) give A179269 (breakdown by width)
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)}
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 10 2019
EXTENSIONS
a(26)-a(42) from Lars Blomberg, May 30 2019
Terms a(43) and beyond from Andrew Howroyd, Aug 27 2019
STATUS
approved