A238874 Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.

1, 0, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 25, 33, 44, 59, 79, 105, 138, 180, 234, 304, 395, 513, 665, 859, 1105, 1416, 1809, 2306, 2935, 3731, 4737, 6005, 7598, 9593, 12085, 15192, 19061, 23875, 29861, 37299, 46532, 57978, 72145, 89650, 111243, 137837, 170545, 210725, 260034, 320492, 394557, 485213, 596074, 731508

Offset: 0

Author

Joerg Arndt, Mar 23 2014

Keywords

• nonn

Examples

			The a(13) = 25 such composition of 13 are:
01:  [ 2 3 8 ]
02:  [ 2 4 7 ]
03:  [ 2 5 6 ]
04:  [ 2 6 5 ]
05:  [ 2 7 4 ]
06:  [ 2 11 ]
07:  [ 3 3 7 ]
08:  [ 3 4 6 ]
09:  [ 3 5 5 ]
10:  [ 3 6 4 ]
11:  [ 3 10 ]
12:  [ 4 3 6 ]
13:  [ 4 4 5 ]
14:  [ 4 5 4 ]
15:  [ 4 9 ]
16:  [ 5 3 5 ]
17:  [ 5 4 4 ]
18:  [ 5 8 ]
19:  [ 6 3 4 ]
20:  [ 6 7 ]
21:  [ 7 6 ]
22:  [ 8 5 ]
23:  [ 9 4 ]
24:  [ 10 3 ]
25:  [ 13 ]
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).

Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).

Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1), j=i..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 24 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 2]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

PARI

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+3)/2) / (1-q)^n );
v=Vec(gf) \\ Joerg Arndt, Mar 30 2014

Formula

G.f.: Sum_{n>=0} q^(n*(n+3)/2) / (1-q)^n. - Joerg Arndt, Mar 30 2014