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A238874
Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.
27
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
0,6
LINKS
Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{n>=0} q^(n*(n+3)/2) / (1-q)^n. - Joerg Arndt, Mar 30 2014
EXAMPLE
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 ]
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 *)
PROG
(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
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).
Sequence in context: A255216 A017836 A321481 * A099559 A247084 A017898
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 23 2014
STATUS
approved