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 A063978 Sum_{i for which n - i*(i-1)/2 >= 0} binomial (n - i*(i-1)/2, i). 4
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A064188. Cf. A219282 (compositions such that p(k)>=k for all k; superdiagonal bargraphs). Sequence in context: A232478 A232476 A132842 * A077868 A109537 A081226 Adjacent sequences: A063975 A063976 A063977 * A063979 A063980 A063981 KEYWORD nonn AUTHOR Helmut Schnitzspan (HSchnitzspan(AT)gmx.de), Sep 05 2001 EXTENSIONS More terms from Dean Hickerson, Sep 06 2001 STATUS approved

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Last modified April 15 18:20 EDT 2024. Contains 371695 sequences. (Running on oeis4.)