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A325546
Number of compositions of n with weakly increasing differences.
13
1, 1, 2, 4, 7, 11, 19, 28, 41, 62, 87, 120, 170, 228, 303, 408, 534, 689, 899, 1145, 1449, 1842, 2306, 2863, 3571, 4398, 5386, 6610, 8039, 9716, 11775, 14157, 16938, 20293, 24166, 28643, 33995, 40134, 47199, 55540, 65088, 75994, 88776, 103328, 119886, 139126
OFFSET
0,3
COMMENTS
Also compositions of n whose plot is concave-up.
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) = 19 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(112) (41) (42)
(211) (113) (51)
(1111) (212) (114)
(311) (123)
(1112) (213)
(2111) (222)
(11111) (312)
(321)
(411)
(1113)
(2112)
(3111)
(11112)
(21111)
(111111)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], LessEqual@@Differences[#]&]], {n, 0, 15}]
PROG
(PARI) \\ Row sums of R(n) give A007294 (=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-w-1)\t + 1, v[i-w-(k-1)*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 28 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 10 2019
EXTENSIONS
More terms from Alois P. Heinz, May 11 2019
STATUS
approved