A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 18, 24, 32, 41, 55, 72, 95, 125, 164, 212, 275, 355, 459, 592, 763, 980, 1257, 1605, 2044, 2598, 3298, 4179, 5290, 6685, 8435, 10623, 13353, 16751, 20978, 26228, 32746, 40831, 50850, 63247, 78569, 97475, 120770, 149429, 184641, 227853, 280832, 345722, 425134, 522232, 640847, 785604

Offset: 0

Author

Joerg Arndt, Mar 24 2014

Keywords

• nonn

Examples

			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 ]
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

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).

Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

Programs

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 *)

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

Formula

G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [Joerg Arndt, Mar 30 2014]