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%I #28 Jan 10 2022 10:44:26
%S 1,2,3,5,8,12,18,27,40,58,83,118,167,235,328,454,624,853,1161,1574,
%T 2125,2856,3821,5090,6754,8931,11773,15474,20280,26502,34533,44870,
%U 58142,75145,96885,124630,159973,204909,261930,334143,425417,540566,685576,867885,1096726,1383545,1742509,2191123,2750980
%N Sum_{i for which n - i*(i-1)/2 >= 0} binomial (n - i*(i-1)/2, i).
%C 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
%H Alois P. Heinz, <a href="/A063978/b063978.txt">Table of n, a(n) for n = 0..1000</a>
%H E. Deutsch, E. Munarini and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%H Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, <a href="https://doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.
%F G.f.: Sum_{k>=1} x^(k*(k-1)/2) / (1-x)^k. - _Vladeta Jovovic_, Sep 25 2004
%e From _Joerg Arndt_, Dec 19 2012: (Start)
%e The a(7) = 18 compositions 7 = p(1) + p(2) + ... + p(m) such that for k>=2 p(k)>=k-1 are
%e [ 1] [ 1 1 2 3 ]
%e [ 2] [ 1 1 5 ]
%e [ 3] [ 1 2 4 ]
%e [ 4] [ 1 3 3 ]
%e [ 5] [ 1 4 2 ]
%e [ 6] [ 1 6 ]
%e [ 7] [ 2 1 4 ]
%e [ 8] [ 2 2 3 ]
%e [ 9] [ 2 3 2 ]
%e [10] [ 2 5 ]
%e [11] [ 3 1 3 ]
%e [12] [ 3 2 2 ]
%e [13] [ 3 4 ]
%e [14] [ 4 1 2 ]
%e [15] [ 4 3 ]
%e [16] [ 5 2 ]
%e [17] [ 6 1 ]
%e [18] [ 7 ]
%e (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1,
%p add(b(n-j, i+1), j=i..n))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..60); # _Alois P. Heinz_, Mar 28 2014
%t Table[ Sum[ Binomial[ n - i(i - 1)/2, i], {i, 0, Floor[ (Sqrt[8n + 1] - 1)/2]} ], {n, 0, 40}]
%o (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
%Y Cf. A064188.
%Y Cf. A219282 (compositions such that p(k)>=k for all k; superdiagonal bargraphs).
%K nonn
%O 0,2
%A Helmut Schnitzspan (HSchnitzspan(AT)gmx.de), Sep 05 2001
%E More terms from _Dean Hickerson_, Sep 06 2001
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