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 A079500 Number of compositions of the integer n in which the first part is >= the other parts. 33
 1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045, 5719, 10780, 20388, 38674, 73562, 140268, 268066, 513350, 984911, 1892875, 3643570, 7023562, 13557020, 26200182, 50691978, 98182666, 190353370, 369393466, 717457656, 1394632365, 2713061899 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Essentially the same as A007059: a(n) = A007059(n+1). In lunar arithmetic in base 2, this is the number of lunar divisors of the number 111...1 (with n 1's). E.g., 1111 has a(4) = 5 divisors (see A048888). - N. J. A. Sloane, Feb 23 2011. First differences of A186537. - N. J. A. Sloane, Feb 23 2011 Number of balanced ordered rooted trees with n non-root nodes (see A048816 for unordered balanced trees); see example. The compositions are obtained from the level sequences by identifying a length-k run of (non-root) levels [t, t+1, t+2, ..., t+k-1] with a part k. - Joerg Arndt, Jul 20 2014 REFERENCES Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295. LINKS T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 400 terms from T. D. Noe) D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing] D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8. Srecko Brlek, Andrea Frosini, Simone Rinaldi, Laurent Vuillon, Tilings by translation: enumeration by a rational language approach, The Electronic Journal of Combinatorics, vol.13, (2006). A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.1. R. Kemp, Balanced ordered trees, Random Structures Algorithms, 5 (1994), pp. 99-121. FORMULA G.f.: (1-z) * sum(k>=0, z^k/(1-2*z+z^(k+1)) ). a(n) = A048888(n) - 1. This is a subsequence of A067399: a(n) = A067399(2^n-1). G.f.: -(1+x^2+ 1/(x-1) )/x*( 1 + x*(x-1)^3*(1-x+x^3)/( Q(0)- x*(x-1)^3*(1-x+x^3)) ), where Q(k) = (x+1)*(2*x-1)*(1-x)^2 + x^(k+2)*(x+x^2+x^3-2*x^4-1 - x^(k+3) + x^(k+5)) - x*(-1+2*x-x^(k+3))*(1-2*x+x^2+x^(k+4)-x^(k+5))*(-1+4*x-5*x^2+2*x^3 - x^(k+2)- x^(k+5) + 2*x^(k+3) - x^(2*k+5) + x^(2*k+6))/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Dec 14 2013 EXAMPLE From Joerg Arndt, Dec 29 2012: (Start) There are a(7)=24 compositions p(1)+p(2)+...+p(m)=7 such that p(k) <= p(1): [ 1]  [ 1 1 1 1 1 1 1 ] [ 2]  [ 2 1 1 1 1 1 ] [ 3]  [ 2 1 1 1 2 ] [ 4]  [ 2 1 1 2 1 ] [ 5]  [ 2 1 2 1 1 ] [ 6]  [ 2 1 2 2 ] [ 7]  [ 2 2 1 1 1 ] [ 8]  [ 2 2 1 2 ] [ 9]  [ 2 2 2 1 ] [10]  [ 3 1 1 1 1 ] [11]  [ 3 1 1 2 ] [12]  [ 3 1 2 1 ] [13]  [ 3 1 3 ] [14]  [ 3 2 1 1 ] [15]  [ 3 2 2 ] [16]  [ 3 3 1 ] [17]  [ 4 1 1 1 ] [18]  [ 4 1 2 ] [19]  [ 4 2 1 ] [20]  [ 4 3 ] [21]  [ 5 1 1 ] [22]  [ 5 2 ] [23]  [ 6 1 ] [24]  [ 7 ] (End) From Joerg Arndt, Jul 20 2014: (Start) The a(7) = 24 balanced ordered rooted trees with 7 non-root nodes are, as level sequences (of the pre-order walk): 01:  [ 0 1 1 1 1 1 1 1 ] 02:  [ 0 1 2 1 2 1 2 2 ] 03:  [ 0 1 2 1 2 2 1 2 ] 04:  [ 0 1 2 1 2 2 2 2 ] 05:  [ 0 1 2 2 1 2 1 2 ] 06:  [ 0 1 2 2 1 2 2 2 ] 07:  [ 0 1 2 2 2 1 2 2 ] 08:  [ 0 1 2 2 2 2 1 2 ] 09:  [ 0 1 2 2 2 2 2 2 ] 10:  [ 0 1 2 3 1 2 3 3 ] 11:  [ 0 1 2 3 2 3 2 3 ] 12:  [ 0 1 2 3 2 3 3 3 ] 13:  [ 0 1 2 3 3 1 2 3 ] 14:  [ 0 1 2 3 3 2 3 3 ] 15:  [ 0 1 2 3 3 3 2 3 ] 16:  [ 0 1 2 3 3 3 3 3 ] 17:  [ 0 1 2 3 4 2 3 4 ] 18:  [ 0 1 2 3 4 3 4 4 ] 19:  [ 0 1 2 3 4 4 3 4 ] 20:  [ 0 1 2 3 4 4 4 4 ] 21:  [ 0 1 2 3 4 5 4 5 ] 22:  [ 0 1 2 3 4 5 5 5 ] 23:  [ 0 1 2 3 4 5 6 6 ] 24:  [ 0 1 2 3 4 5 6 7 ] (End) From Gus Wiseman, Oct 07 2018: (Start) The a(0) = 1 through a(6) = 14 balanced rooted plane trees:   o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)           ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))                  (((o)))  (((oo)))   (((ooo)))    (((oooo)))                           ((o)(o))   ((o)(oo))    ((o)(ooo))                           ((((o))))  ((oo)(o))    ((oo)(oo))                                      ((((oo))))   ((ooo)(o))                                      (((o)(o)))   ((((ooo))))                                      (((((o)))))  (((o)(oo)))                                                   (((oo)(o)))                                                   ((o)(o)(o))                                                   (((((oo)))))                                                   ((((o)(o))))                                                   (((o))((o)))                                                   ((((((o)))))) (End) MAPLE M:=101: t1:=add( (1-x)*x^k/(1-2*x+x^k), k=1..M): series(t1, x, M-1); seriestolist(%); # second Maple program: b:= proc(n, m) option remember; `if`(n=0, 1,       `if`(m=0, add(b(n-j, j), j=1..n),       add(b(n-j, min(n-j, m)), j=1..min(n, m))))     end: a:= n-> b(n, 0): seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014 MATHEMATICA nn=36; CoefficientList[Series[Sum[x^i/(1-(x-x^(i+1))/(1-x)), {i, 0, nn}], {x, 0, nn}], x]  (* Geoffrey Critzer, Mar 12 2013 *) b[n_, m_] := b[n, m] = If[n==0, 1, If[m==0, Sum[b[n-j, j], {j, 1, n}], Sum[ b[n-j, Min[n-j, m]], {j, 1, Min[n, m]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *) CROSSREFS Cf. A188541. Cf. A000081 A000311, A001678, A120803, A320154, A320160, A316624, A320169. Sequence in context: A108351 A038495 A007059 * A108296 A072100 A290075 Adjacent sequences:  A079497 A079498 A079499 * A079501 A079502 A079503 KEYWORD nonn AUTHOR Arnold Knopfmacher, Jan 21 2003 EXTENSIONS Offset corrected by N. J. A. Sloane, Feb 23 2011 More terms from N. J. A. Sloane, Feb 24 2011 Further edits (required in order to clarify the definition - is the first part >= the rest. or only > the rest? Answer: the former; for the latter, see A007059) by N. J. A. Sloane, May 08 2011 STATUS approved

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Last modified November 19 00:12 EST 2018. Contains 317332 sequences. (Running on oeis4.)