OFFSET
0,2
COMMENTS
Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k)>=k-1 for k>=2, see example. - Joerg Arndt, Dec 19 2012
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.
FORMULA
G.f.: Sum_{k>=1} x^(k*(k-1)/2) / (1-x)^k. - Vladeta Jovovic, Sep 25 2004
EXAMPLE
From Joerg Arndt, Dec 19 2012: (Start)
The a(7) = 18 compositions 7 = p(1) + p(2) + ... + p(m) such that for k>=2 p(k)>=k-1 are
[ 1] [ 1 1 2 3 ]
[ 2] [ 1 1 5 ]
[ 3] [ 1 2 4 ]
[ 4] [ 1 3 3 ]
[ 5] [ 1 4 2 ]
[ 6] [ 1 6 ]
[ 7] [ 2 1 4 ]
[ 8] [ 2 2 3 ]
[ 9] [ 2 3 2 ]
[10] [ 2 5 ]
[11] [ 3 1 3 ]
[12] [ 3 2 2 ]
[13] [ 3 4 ]
[14] [ 4 1 2 ]
[15] [ 4 3 ]
[16] [ 5 2 ]
[17] [ 6 1 ]
[18] [ 7 ]
(End)
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, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 28 2014
MATHEMATICA
Table[ Sum[ Binomial[ n - i(i - 1)/2, i], {i, 0, Floor[ (Sqrt[8n + 1] - 1)/2]} ], {n, 0, 40}]
PROG
(PARI) N=66; q='q+O('q^N); Vec( sum(n=1, N, q^(n*(n-1)/2) / (1-q)^n ) ) \\ Joerg Arndt, Mar 30 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Helmut Schnitzspan (HSchnitzspan(AT)gmx.de), Sep 05 2001
EXTENSIONS
More terms from Dean Hickerson, Sep 06 2001
STATUS
approved