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).
LINKS
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