OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)
FORMULA
G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [Joerg Arndt, Mar 30 2014]
EXAMPLE
There are a(12) = 24 such compositions of 12:
01: [ 1 2 3 6 ]
02: [ 1 2 4 5 ]
03: [ 1 2 5 4 ]
04: [ 1 2 9 ]
05: [ 1 3 3 5 ]
06: [ 1 3 4 4 ]
07: [ 1 3 8 ]
08: [ 1 4 3 4 ]
09: [ 1 4 7 ]
10: [ 1 5 6 ]
11: [ 1 6 5 ]
12: [ 1 7 4 ]
13: [ 1 8 3 ]
14: [ 1 11 ]
15: [ 2 3 7 ]
16: [ 2 4 6 ]
17: [ 2 5 5 ]
18: [ 2 6 4 ]
19: [ 2 10 ]
20: [ 3 4 5 ]
21: [ 3 9 ]
22: [ 4 8 ]
23: [ 5 7 ]
24: [ 12 ]
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i=0, add(b(n-j, j+1), j=1..n),
add(b(n-j, i+1), j=i..n)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 26 2014
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, i, n}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
PROG
(PARI) N=66; q='q+O('q^N);
gf = 1 + sum(n=1, N, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) );
v=Vec(gf) \\ Joerg Arndt, Mar 30 2014
CROSSREFS
Cf. A238860 (partitions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 24 2014
STATUS
approved