A063978 Sum_{i for which n - i*(i-1)/2 >= 0} binomial (n - i*(i-1)/2, i).

1, 2, 3, 5, 8, 12, 18, 27, 40, 58, 83, 118, 167, 235, 328, 454, 624, 853, 1161, 1574, 2125, 2856, 3821, 5090, 6754, 8931, 11773, 15474, 20280, 26502, 34533, 44870, 58142, 75145, 96885, 124630, 159973, 204909, 261930, 334143, 425417, 540566, 685576, 867885, 1096726, 1383545, 1742509, 2191123, 2750980

Offset: 0

Author

Helmut Schnitzspan (HSchnitzspan(AT)gmx.de), Sep 05 2001

Keywords

• nonn

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

Examples

			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)
		

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.

Crossrefs

Cf. A064188.

Cf. A219282 (compositions such that p(k)>=k for all k; superdiagonal bargraphs).

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, 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}]

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

Formula

G.f.: Sum_{k>=1} x^(k*(k-1)/2) / (1-x)^k. - Vladeta Jovovic, Sep 25 2004

Extensions

More terms from Dean Hickerson, Sep 06 2001