A186537 -id:A186537 - cp's OEIS frontend

A079500 Number of compositions of the integer n in which the first part is >= the other parts.

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

Offset: 0

Arnold Knopfmacher, Jan 21 2003

nonn

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

			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)

Arnold Knopfmacher and Neville Robbins, Compositions with parts constrained by the leading summand, Ars Combin. 76 (2005), 287-295.

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (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, and 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.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A188541.
Cf. A000081 A000311, A001678, A120803, A320154, A320160, A316624, A320169.
Cf. A092921.

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

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 *)

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
a(n) = Sum_{j=1..n} F(j, n+1-j), where F(n,k) is the n-th k-generalized Fibonacci number A092921(k,n). - Gregory L. Simay, Aug 21 2022

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

A368279 a(n) is the number of compositions of n where the first part is the largest part and the last part is not 1. Row sums of A368579.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 10, 19, 34, 63, 116, 216, 402, 754, 1417, 2674, 5061, 9608, 18286, 34888, 66706, 127798, 245284, 471561, 907964, 1750695, 3379992, 6533458, 12643162, 24491796, 47490688, 92170704, 179040096, 348064190, 677174709, 1318429534, 2568691317

Offset: 0

Peter Luschny, Jan 04 2024

nonn

Considering more generally the family of generating functions (1 - x)^n * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)) one finds several sequences related to compositions as indicated in the cross-references.

The compositions considered here can also be understood as perfectly balanced, ordered trees. See the linked illustrations. - Peter Luschny, Feb 26 2024

			a(0) = card({[0]}) = 1.
a(1) = card({}) = 0.
a(2) = card({[2]}) = 1.
a(3) = card({[3]}) = 1.
a(4) = card({[2, 2], [4]}) = 2.
a(5) = card({[2, 1, 2], [3, 2], [5]}) = 3.
a(6) = card({[2, 2, 2], [2, 1, 1, 2], [3, 3], [3, 1, 2], [4, 2], [6]}) = 6.
a(7) = card({[2, 2, 1, 2], [2, 1, 2, 2], [2, 1, 1, 1, 2], [3, 2, 2], [3, 1, 3], [3, 1, 1, 2], [4, 3], [4, 1, 2], [5, 2], [7]}) = 10.
a(8) = card({[2, 2, 2, 2],  [2, 2, 1, 1, 2], [2, 1, 2, 1, 2], [2, 1, 1, 2, 2], [2, 1, 1, 1, 1, 2], [3, 3, 2], [3, 2, 3], [3, 2, 1, 2], [3, 1, 2, 2], [3, 1, 1, 3], [3, 1, 1, 1, 2], [4, 4], [4, 2, 2], [4, 1, 3], [4, 1, 1, 2], [5, 3], [5, 1, 2], [6, 2], [8]}) = 19.

Peter Luschny, A generator for the A368279 compositions.
Peter Luschny, Some perfectly balanced, ordered trees illustrating A368279.

Cf. A368579, A369492, A007059, A092921, A188541.

Cf. A369115 (n=-2), A186537 left shifted (n=-1), A079500 (n=0), this sequence (n=1), A369116 (n=2).

gf := (1 - x)*sum(x^j / (1 - sum(x^k, k = 1..j)), j = 0..42):
ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..37);
# Peter Luschny, Jan 19 2024

from functools import cache
@cache
def F(k, n):
    return sum(F(k,n-j) for j in range(1,min(k,n))) if n>1 else n
def a(n): return sum(F(k+1, n+1-k) - F(k+1, n-k) for k in range(n+1))
print([a(n) for n in range(38)])

def C(n): return sum(Compositions(n, max_part=k, inner=[k]).cardinality()
                 for k in (0..n))
def a(n): return C(n) - C(n-1) if n > 1 else 1 - n
print([a(n) for n in (0..28)])

a(n) = Sum_{k=0..n} (F(k+1, n+1-k) - F(k+1, n-k)) where F(k, n) = Sum_{j=1..min(k, n)} F(k, n-j) if n > 1 and otherwise n. F(k, n) refers to the generalized Fibonacci number A092921.
a(n) = A007059(n+1) - A007059(n).
G.f.: (1 - x)*(Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k ))) = (1 - x) * GfA079500. - Peter Luschny, Jan 20 2024

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A079500 Number of compositions of the integer n in which the first part is >= the other parts.

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A368279 a(n) is the number of compositions of n where the first part is the largest part and the last part is not 1. Row sums of A368579.

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A369115 Expansion of (1 - x)^(-2) * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

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A369116 Expansion of (1 - x)^2 * Sum_{j>=0} (x^j / (1 - Sum_{k=1..j} x^k)).

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