OFFSET
0,5
COMMENTS
We note that the definition implies that the maximum part of the composition must be strictly greater than 1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..800
FORMULA
a(n) = Sum_{i>=2} x^(i+1)/(1-Sum_{j=2..i-1} x^j)/(1 - Sum{k=1..i-1} x^k)/(1 - Sum_{m=1..i} x^m).
a(n) ~ 2^(n-1). - Vaclav Kotesovec, May 01 2014
EXAMPLE
a(5) = 7 because we have: 1+4, 1+1+3, 1+2+2, 1+3+1, 1+1+1+2, 1+1+2+1, 1+2+1+1.
MAPLE
b:= proc(n, t, m) option remember;
`if`(n=0, t, add(`if`(j=1 and m>1, b(n-1, 1, m),
`if`(j>=m, b(n-j, 0, j), b(n-j, t, m))), j=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..45); # Alois P. Heinz, Feb 17 2014
MATHEMATICA
nn=30; CoefficientList[Series[Sum[x^(i+1)/(1-Sum[x^j, {j, 2, i-1}])/(1-Sum[x^k, {k, 1, i-1}])/(1-Sum[x^m, {m, 1, i}]), {i, 2, nn}], {x, 0, nn}], x]
(* or *)
Table[Length[Select[Level[Table[Select[Compositions[n, k], Count[#, 0]==0&], {k, 1, n}], {2}], Min[Position[#, 1]]<Min[Position[#, Max[#]]]&]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Feb 17 2014
STATUS
approved