A156041 -id:A156041 - cp's OEIS frontend

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

1, 1, 3, 4, 6, 8, 11, 13, 17, 20, 24, 28, 33, 37, 43, 48, 54, 60, 67, 73, 81, 88, 96, 104, 113, 121, 131, 140, 150, 160, 171, 181, 193, 204, 216, 228, 241, 253, 267, 280, 294, 308, 323, 337, 353, 368, 384, 400, 417, 433, 451, 468, 486, 504, 523, 541, 561, 580, 600

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

For n = 1, 2 these are just the triangular numbers. a(n) is always at least 1/3 of the corresponding triangular number, since each partition of this type gives up to three ordered partitions with the same cyclical order.
An alternative definition, which avoids using parts of size 0: a(n) is the third diagonal of A184957. - N. J. A. Sloane, Feb 27 2011
Diagonal sums of the triangle formed by rows T(2, k) k = 0, 1, ..., 2m of ascending m-nomial triangles (see A004737):
1
1 2 1
1 2 3 2 1
1 2 3 4 3 2 1
1 2 3 4 5 4 3 2 1
1 2 3 4 5 6 5 4 3 2 1
- Bob Selcoe, Feb 07 2014
Arrange A004396 in rows successively shifted to the right two spaces and sum the columns:
1  1  2  3  3  4  5  5  6 ...
      1  1  2  3  3  4  5 ...
            1  1  2  3  3 ...
                  1  1  2 ...
                        1 ...
------------------------------
1  1  3  4  6  8 11 13 17 ... - L. Edson Jeffery, Jul 30 2014
a(n) is the dimension of three-dimensional (2n + 2)-homogeneous polynomial vector fields with full tetrahedral symmetry (for a given orthogonal representation), and which are solenoidal. - Giedrius Alkauskas, Sep 30 2017
Also the number of compositions of n + 3 into three parts, the first at least as great as each of the other two. Also the number of compositions of n + 4 into three parts, the first strictly greater than each of the other two. - Gus Wiseman, Oct 09 2020

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 11*x^6 + 13*x^7 + 17*x^8 + 20*x^9 + ...
The a(4) = 6 compositions of 4 are: (4 0 0), (3 1 0), (3 0 1), (2 2 0), (2 1 1), (2 0 2).
From _Gus Wiseman_, Oct 05 2020: (Start)
The a(0) = 1 through a(7) = 13 triples of nonnegative integers summing to n where the first is at least as great as each of the other two are:
  (000)  (100)  (101)  (111)  (202)  (212)  (222)  (313)
                (110)  (201)  (211)  (221)  (303)  (322)
                (200)  (210)  (220)  (302)  (312)  (331)
                       (300)  (301)  (311)  (321)  (403)
                              (310)  (320)  (330)  (412)
                              (400)  (401)  (402)  (421)
                                     (410)  (411)  (430)
                                     (500)  (420)  (502)
                                            (501)  (511)
                                            (510)  (520)
                                            (600)  (601)
                                                   (610)
                                                   (700)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG] (2017), Section 4.3.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

For compositions into 4 summands see A156039; also see A156041 and A156042.
Cf. A184957, A071619 (bisection).
A001399(n-2)*2 is the strict case.
A001840(n-2) is the version with opposite relations.
A001840(n-1) is the version with strict opposite relations.
A069905 is the case with strict relations.
A014311 ranks 3-part compositions, with strict case A337453.
A014612 ranks 3-part partitions, with strict case A007304.
Cf. A000217, A001523, A211540, A218004, A220377, A337483, A337484.

a:= proc(n) local m, r; m := iquo(n, 6, 'r'); (4 +6*m +2*r) *m + [1, 1, 3, 4, 6, 8][r+1] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

nn = 58; CoefficientList[Series[x^3/(1 - x^2)^2/(1 - x^3) + 1/(1 - x^2)^2/(1 - x), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 14 2013 *)
CoefficientList[Series[(1 + x^2)/((1 + x) * (1 + x + x^2) * (1 - x)^3), {x, 0, 58}], x] (* L. Edson Jeffery, Jul 29 2014 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 1, 3, 4, 6, 8}, 60] (* Harvey P. Dale, May 28 2015 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n+3,{3}],#[[1]]>=#[[2]]&&#[[1]]>=#[[3]]&]],{n,0,15}] (* Gus Wiseman, Oct 05 2020*)

{a(n) = n*(n+4)\6 + 1}; /* Michael Somos, Mar 26 2017 */

G.f.: (x^2+1) / (1-x-x^2+x^4+x^5-x^6). - Alois P. Heinz, Jun 14 2009
Slightly nicer g.f.: (1+x^2)/((1-x)*(1-x^2)*(1-x^3)). - N. J. A. Sloane, Apr 29 2011
a(n) = A007590(n+2) - A000212(n+2). - Richard R. Forberg, Dec 08 2013
a(2*n) = A071619(n+1). - L. Edson Jeffery, Jul 29 2014
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 6, a(5) = 8. - Harvey P. Dale, May 28 2015
a(n) = (n^2 + 4*n + 3)/6 + IF(MOD(n, 2) = 0, 1/2) + IF(MOD(n, 3) = 1, -1/3). - Heinrich Ludwig, Mar 21 2017
a(n) = 1 + floor((n^2 + 4*n)/6). - Giovanni Resta, Mar 21 2017
Euler transform of length 4 sequence [1, 2, 1, -1]. - Michael Somos, Mar 26 2017
a(n) = a(-4 - n) for all n in Z. - Michael Somos, Mar 26 2017
0 = a(n)*(-1 + a(n) - 2*a(n+1) - 2*a(n+2) + 2*a(n+3)) + a(n+1)*(+1 + a(n+1) + 2*a(n+2) - 2*a(n+3)) + a(n+2)*(+1 + a(n+2) - 2*a(n+3)) + a(n+3)*(-1 + a(n+3)) for all n in Z. - Michael Somos, Mar 26 2017
a(n) = round((n+1)*(n+3)/6). - Bill McEachen, Feb 16 2021
Sum_{n>=0} 1/a(n) = 3/2 + Pi^2/36 + (tan(c1)-1)*c1 + 3*c2*sinh(c2)/(1+2*cosh(c2)), where c1 = Pi/(2*sqrt(3)) and c2 = Pi*sqrt(2)/3. - Amiram Eldar, Dec 10 2022
E.g.f.: ((16 + 15*x + 3*x^2)*cosh(x) + 2*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (7 + 15*x + 3*x^2)*sinh(x))/18. - Stefano Spezia, Apr 05 2023

More terms from Alois P. Heinz, Jun 14 2009

A156039 Number of compositions (ordered partitions) of n into 4 parts, where the first is at least as great as each of the others.

Original entry on oeis.org

1, 1, 4, 7, 11, 17, 26, 35, 48, 63, 81, 102, 127, 154, 187, 223, 263, 308, 359, 413, 474, 540, 612, 690, 775, 865, 964, 1069, 1181, 1301, 1430, 1565, 1710, 1863, 2025, 2196, 2377, 2566, 2767, 2977, 3197, 3428, 3671, 3923, 4188, 4464, 4752, 5052, 5365, 5689

Offset: 0

Jack W Grahl, Feb 02 2009, Feb 11 2009

nonn

For n=1,2 these are just the tetrahedral numbers. a(n) is always at least 1/4 of the corresponding tetrahedral number, since each partition of this type gives up to four ordered partitions with the same cyclical order.

Diagonal sums of the irregular triangle A109439, for example a(0)=1, a(1)=1, a(2)=1+3, a(3)=1+3+3, a(4)=1+3+6+1. - Bob Selcoe, Feb 09 2014

			For n = 3 the a(3) = 7 compositions are: (3 0 0 0) (2 1 0 0) (2 0 1 0) (2 0 0 1) (1 1 1 0) (1 1 0 1) (1 0 1 1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -1, -1, 1, -1, 2, -1).

For partitions into 3 summands see A156040; also see A156041 and A156042.

a:= proc(n) local m, r; m:= iquo(n, 12, 'r'); r:= r+1; (9 +(27 +72*m +18*r)*m +((9 +3*r) *r-12) /2)*m +[1, 1, 4, 7, 11, 17, 26, 35, 48, 63, 81, 102][r] end: seq(a(n), n=0..60); # Alois P. Heinz, Jun 14 2009

LinearRecurrence[{2, -1, 1, -1, -1, 1, -1, 2, -1}, {1, 1, 4, 7, 11, 17, 26, 35, 48}, 60] (* Jean-François Alcover, May 17 2018 *)

G.f.: ( 1-x+3*x^2-x^3+x^4 ) / ( (1+x)*(1+x^2)*(1+x+x^2)*(x-1)^4 ). - Alois P. Heinz, Jun 14 2009

More terms from Alois P. Heinz, Jun 14 2009

A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 3, 6, 11, 1, 3, 8, 17, 32, 1, 4, 11, 26, 54, 102, 1, 4, 13, 35, 82, 172, 331, 1, 5, 17, 48, 120, 272, 567, 1101, 1, 5, 20, 63, 170, 412, 918, 1906, 3724, 1, 6, 24, 81, 235, 607, 1434, 3152, 6518, 12782, 1, 6, 28, 102, 317, 872, 2180, 5049, 10978, 22616, 44444

Offset: 1

Jack W Grahl, Feb 02 2009

The value is smaller than the number of compositions (ordered partitions) of n into k parts and at least the number of (unordered) partitions.

			A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
[5,0,0], [4,1,0], [4,0,1], [3,2,0], [3,0,2], [3,1,1], [2,2,1], [2,1,2].
1
1  2
1  2  4
1  3  6  11
1  3  8  17  32
1  4  11 26  54  102

Alois P. Heinz, Rows n = 1..100, flattened

A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.

b:= proc(n,i,m) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n<=m, 1, 0)
    else add(b(n-k, i-1, m), k=0..m)
      fi
    end:
A:= (n,k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
seq(seq(A(n,k), k=1..n), n=1..12); # Alois P. Heinz, Jun 14 2009

nn=10; Table[Table[Coefficient[Series[Sum[x^i((1-x^(i+1))/(1-x))^(k-1), {i, 0, n}], {x, 0, nn}], x^n], {k, 1, n}], {n, 1, nn}]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

More terms from Alois P. Heinz, Jun 14 2009

A156043 A(n,n), where A(n,k) is the number of compositions (ordered partitions) of n into k parts (parts of size 0 being allowed), with the first part being greater than or equal to all the rest.

Original entry on oeis.org

1, 2, 4, 11, 32, 102, 331, 1101, 3724, 12782, 44444, 156334, 555531, 1991784, 7197369, 26186491, 95847772, 352670170, 1303661995, 4838822931, 18025920971, 67371021603, 252538273442, 949164364575, 3576145084531, 13503991775252

Offset: 1

Jack W Grahl, Feb 02 2009

nonn

The value is smaller than the number of compositions of n into k parts and at least the number of (unordered) partitions.
It is also at least the number of compositions of n into n parts  divided by n. From these bounds: C(2*n-1,n-1)/n <= a(n) <= C(2*n-1,n-1). - Robert Gerbicz, Apr 06 2011
a(n) is also the number of Dyck paths of semilength 2n such that each level has exactly n peaks or no peaks. a(3) = 4: //\\//\\//\\, ///\\//\/\\\, ///\/\\//\\\, ////\/\/\\\\. - Alois P. Heinz, Jun 04 2017

			a(4) = 11: the 11 compositions of this type of 4 into 4 parts being
(4,0,0,0); (3,1,0,0); (3,0,1,0); (3,0,0,1);
(2,2,0,0); (2,0,2,0); (2,0,0,2); (2,1,1,0);
(2,1,0,1); (2,0,1,1); (1,1,1,1)

Robert Gerbicz, Table of n, a(n) for n = 1..500

A156041 gives the full array A(n, k). See also A156039, A156040 and A156042.

One of two bisections of A188624 (see also A188625).

b:= proc(n,i,m) option remember; if n<0 then 0 elif n=0 then 1 elif i=1 then `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi end: A:= (n,k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n): seq(A(n,n), n=1..30); # Alois P. Heinz, Jun 14 2009

b[n_, i_, m_] := b[n, i, m] = Which[n<0, 0, n==0, 1, i==1, If[n <= m, 1, 0], True, Sum[b[n-k, i-1, m], {k, 0, m}]]; A[n_, k_] := Sum[b[n-m, k-1, m], {m, Ceiling[n/k], n}]; Table[A[n, n], {n, 1, 30}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

N=120;v=vector(N,i,0);for(d=1,N,A=matrix(N,N,i,j,0);A[1,1]=1; for(i=1,N-1,for(j=0,N-1,s=0;for(k=0,min(j,d), s+=A[i,j-k+1]);A[i+1,j+1]=s)); for(i=d,N,v[i]+=A[i,i-d+1]));for(i=1,N,print1(v[i]", ")) \\ Robert Gerbicz, Apr 06 2011

More terms from Alois P. Heinz, Jun 14 2009

Edited by N. J. A. Sloane, Apr 06 2011

A184957 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) is the number of compositions of n into k parts the first of which is >= all the other parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 4, 1, 1, 1, 3, 6, 7, 5, 1, 1, 1, 4, 8, 11, 11, 6, 1, 1, 1, 4, 11, 17, 19, 16, 7, 1, 1, 1, 5, 13, 26, 32, 31, 22, 8, 1, 1, 1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1, 1, 6, 20, 48, 82, 102, 93, 71, 37, 10, 1, 1, 1, 6, 24, 63, 120, 172, 180, 148, 101, 46, 11, 1, 1, 1, 7, 28, 81, 170, 272, 331, 302, 227, 139, 56, 12, 1, 1

Offset: 1

N. J. A. Sloane, Feb 27 2011

If the final diagonal is omitted, this gives the triangular array visible in A156041 and A186807.

			Triangle begins:
  [1],
  [1, 1],
  [1, 1, 1],
  [1, 2, 1, 1],
  [1, 2, 3, 1, 1],
  [1, 3, 4, 4, 1, 1],
  [1, 3, 6, 7, 5, 1, 1],
  [1, 4, 8, 11, 11, 6, 1, 1],
  [1, 4, 11, 17, 19, 16, 7, 1, 1],
  [1, 5, 13, 26, 32, 31, 22, 8, 1, 1],
  [1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1],
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows 1 <= n <= 50, flattened).
Gal Gross, Maximally additively reducible subsets of the integers, arXiv:1908.05220 [math.CO], 2019.

Cf. A156040, A156041, A186807, A079500 (row sums).

# The following Maple program is a modification of Alois P. Heinz's program for A156041
b:= proc(n, i, m) option remember;
       if n<0 then 0 elif n=0 then 1 elif i=1 then
      `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi
    end:
A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
[seq([seq(A(d-k, k), k=1..d)], d=1..14)];

Map[Select[#,#>0&]&,Drop[nn=11;CoefficientList[Series[Sum[x^i/(1-y(x-x^(i+1))/(1-x)),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

T(n,k) = A156041(n-k,k).

O.g.f.: Sum_{i>=1} x^i/(1 - y*(x - x^(i+1))/(1-x)). - Geoffrey Critzer, Jul 15 2013

A186807 Triangle read by rows: T(n,p) (n >= 2, 1 <= p <= n-1) = number of compositions of n into p parts, with first part >= all other parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 3, 1, 1, 5, 7, 6, 3, 1, 1, 6, 11, 11, 8, 4, 1, 1, 7, 16, 19, 17, 11, 4, 1, 1, 8, 22, 31, 32, 26, 13, 5, 1, 1, 9, 29, 48, 56, 54, 35, 17, 5, 1, 1, 10, 37, 71, 93, 102, 82, 48, 20, 6, 1, 1, 11, 46, 101, 148, 180, 172, 120, 63, 24, 6, 1

Offset: 2

N. J. A. Sloane, Feb 26 2011

This triangle arose in connection with a problem involving "lunar arithmetic".

Take the triangle formed by the antidiagonals of A156041 and reverse each row.

			Triangle begins:
1,
1, 1,
1, 2, 1,
1, 3, 2, 1,
1, 4, 4, 3, 1,
1, 5, 7, 6, 3, 1,
1, 6, 11, 11, 8, 4, 1,
1, 7, 16, 19, 17, 11, 4, 1,
1, 8, 22, 31, 32, 26, 13, 5, 1,
1, 9, 29, 48, 56, 54, 35, 17, 5, 1,
...

D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A156041. This is also A184957 with the last diagonal omitted.

cp's OEIS Frontend

A156040 Number of compositions (ordered partitions) of n into 3 parts (some of which may be zero), where the first is at least as great as each of the others.

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A156039 Number of compositions (ordered partitions) of n into 4 parts, where the first is at least as great as each of the others.

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A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.

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A156043 A(n,n), where A(n,k) is the number of compositions (ordered partitions) of n into k parts (parts of size 0 being allowed), with the first part being greater than or equal to all the rest.

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A184957 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) is the number of compositions of n into k parts the first of which is >= all the other parts.

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A186807 Triangle read by rows: T(n,p) (n >= 2, 1 <= p <= n-1) = number of compositions of n into p parts, with first part >= all other parts.

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A188624 a(n) = greatest entry in row n of A184957.

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