A079500 -id:A079500 - cp's OEIS frontend

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.

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

A186537 G.f.: Sum( x^k/(1-2*x+x^k), k=1..oo).

Original entry on oeis.org

0, 1, 2, 4, 7, 12, 20, 34, 58, 101, 178, 318, 574, 1046, 1920, 3548, 6593, 12312, 23092, 43480, 82154, 155716, 295984, 564050, 1077400, 2062311, 3955186, 7598756, 14622318, 28179338, 54379520, 105071498, 203254164, 393607534, 763001000, 1480458656, 2875091021, 5588152920, 10869906136

Offset: 0

N. J. A. Sloane, Feb 23 2011

nonn

This arose while studying the properties of A079500.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 257 terms from N. J. A. Sloane)

First differences give A079500.

add( x^k/(1-2*x+x^k), k=1..61); series(%,x,60); 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:= proc(n) a(n):= `if`(n=0, 0, b(n-1, 0)+a(n-1)) end:
seq(a(n), n=0..40);  # Alois P. Heinz, May 01 2014

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_] := If[n == 0, 0, b[n-1, 0] + a[n-1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 05 2014, after Alois P. Heinz *)

G.f.: -(1+x^2+ 1/(x-1) )/(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

A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 19, 25, 35, 46, 65, 88, 124, 171, 242, 334, 470, 653, 921, 1287, 1822, 2565, 3640, 5144, 7311, 10360, 14734, 20918, 29781, 42361, 60389, 86069, 122893, 175479, 250922, 358863

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

An unlabeled rooted tree is semi-binary if all out-degrees are <= 2, and balanced if all leaves are the same distance from the root. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			The a(1) = 1 through a(7) = 6 balanced semi-binary rooted trees:
  o  (o)  (oo)   ((oo))   (((oo)))   ((o)(oo))    ((oo)(oo))
          ((o))  (((o)))  ((o)(o))   ((((oo))))   (((o)(oo)))
                          ((((o))))  (((o)(o)))   (((((oo)))))
                                     (((((o)))))  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Gus Wiseman, The a(13) = 35 balanced semi-binary rooted trees.
Gus Wiseman, The a(15) = 65 balanced semi-binary rooted trees.
Gus Wiseman, The a(16) = 88 balanced semi-binary rooted trees.
Gus Wiseman, The a(18) = 171 balanced semi-binary rooted trees.

Cf. A001190, A048816, A079500, A111299, A292050, A298204, A301345, A320271.

saur[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[saur/@ptn]],SameQ@@Length/@Position[#,{}]&],{ptn,Select[IntegerPartitions[n-1],Length[#]<=2&]}]];
Table[Length[saur[n]],{n,20}]

A096202 Number of coverings of {1...n} by translation of a single set.

Original entry on oeis.org

1, 2, 3, 6, 11, 22, 45, 92, 188, 382, 791, 1632, 3357, 6922, 14289, 29542, 61013, 126142, 260664, 538850, 1113372, 2300954, 4752279, 9814226, 20257082, 41798206, 86204773, 177729712, 366231907, 754356336, 1553063269, 3196028942, 6573883225, 13515943986, 27775807554

Offset: 1

Jon Wild, Jul 27 2004

nonn

The number of sets (up to translation) that with their translations can cover {1...n} in at least one way is given by A079500(n). For example, for n = 5 the 8 sets are {1}, {1,2}, {1,3}, {1,2,3}, {1,2,4}, {1,3,4}, {1,2,3,4}, {1,2,3,4,5}. - Andrew Howroyd, Nov 06 2019

			a(5)=11 because the following are the 11 coverings of {1...5}, each one of which only uses a single set and its translations:
   {{1},{2},{3},{4},{5}}
   {{1,2},{3,4},{4,5}}
   {{1,2},{2,3},{3,4},{4,5}}
   {{1,2},{2,3},{4,5}}
   {{1,3},{2,4},{3,5}}
   {{1,2,3},{2,3,4},{3,4,5}}
   {{1,2,3},{3,4,5}}
   {{1,2,4},{2,3,5}}
   {{1,3,4},{2,4,5}}
   {{1,2,3,4},{2,3,4,5}}
   {{1,2,3,4,5}}

Cf. A079500, A096154, A096203.

covers(all, v)={
  my(u=vector(#v+1)); for(i=1, #v, u[i+1]=bitor(u[i], v[i]));
  my(recurse(k,b) = if(bitnegimply(b,u[k+1]), 0, if(k==0, 1, my(t=bitnegimply(b,v[k])); if(t==b, 2*self()(k-1, b), self()(k-1, b) + self()(k-1, t)) )));
  recurse(#v, all)
}
a(n)={sum(i=2^(n-1), 2^n-1, covers(2^n-1, vector(valuation(i,2)+1, j, i>>(j-1))))} \\ Andrew Howroyd, Nov 06 2019

a(14)-a(32) from Andrew Howroyd, Nov 06 2019

a(33)-a(35) from Jinyuan Wang, Jun 09 2021

A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 9, 19, 38, 79, 163, 352, 750, 1633, 3558, 7783, 17020, 37338, 81920, 180399, 398600, 885101, 1975638, 4435741, 10013855, 22726109, 51807432, 118545425, 272024659, 625488420, 1440067761, 3317675261, 7644488052, 17610215982, 40547552277, 93298838972, 214516498359, 492844378878

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 19 rooted identity trees:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             ((1)(2))   (211)       (221)
             ((1)(11))  (1111)      (311)
                        ((1)(3))    (2111)
                        ((1)(21))   (11111)
                        ((2)(11))   ((1)(4))
                        ((1)(111))  ((2)(3))
                                    ((1)(31))
                                    ((1)(22))
                                    ((2)(21))
                                    ((3)(11))
                                    ((1)(211))
                                    ((11)(21))
                                    ((2)(111))
                                    ((1)(1111))
                                    ((11)(111))
                                    ((1)(2)(11))

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A292504, A300660, A319312.

Cf. A320154, A320155, A320160, A320171, A320173, A320177, A320178, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[gig[y],SameQ@@Length/@Position[#,_Integer]&]],{y,Sort /@IntegerPartitions[n]}],{n,8}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A382312 Irregular triangle read by rows: T(n,k) is the number of compositions of n with k records.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 1, 0, 5, 3, 0, 8, 8, 0, 14, 17, 1, 0, 24, 36, 4, 0, 43, 72, 13, 0, 77, 143, 36, 0, 140, 281, 90, 1, 0, 256, 550, 213, 5, 0, 472, 1073, 484, 19, 0, 874, 2093, 1068, 61, 0, 1628, 4079, 2308, 177, 0, 3045, 7950, 4912, 476, 1, 0, 5719, 15498, 10328, 1217, 6

Offset: 0

John Tyler Rascoe, Mar 21 2025

A record in a composition is a part that is greater than all parts before it, reading left to right. The first part of any nonempty composition is considered a record.

			Triangle begins:
    k=0    1    2   3  4
 n= 0 1;
 n= 1 0,   1;
 n= 2 0,   2;
 n= 3 0,   3,   1;
 n= 4 0,   5,   3;
 n= 5 0,   8,   8;
 n= 6 0,  14,  17,  1;
 n= 7 0,  24,  36,  4;
 n= 8 0,  43,  72, 13;
 n= 9 0,  77, 143, 36;
 n=10 0, 140, 281, 90, 1;
 ...
The composition (2,1,1,2,4,2,1,5,7) has 4 records.
                 ^       ^     ^ ^
T(4,1) = 5 counts: (4), (3,1), (2,2), (2,1,1), (1,1,1,1).
T(4,2) = 3 counts: (1,1,2), (1,2,1), (1,1,3).

Alois P. Heinz, Rows n = 0..500, flattened (first 201 rows from John Tyler Rascoe)

Cf. A002024 (row lengths), A011782 (row sums), A079500 (column k=1), A336482, A352525.

b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
      b(n-j, max(m, j))*`if`(j>m, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..16);  # Alois P. Heinz, Mar 28 2025

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h=prod(i=1,N,1+y*x^i*(1-x)/(1-2*x+x^(i+1)))); vector(N, n, Vecrev(polcoeff(h, n-1)))}
T_xy(12)

G.f.: Product_{i>0} (1 + y*x^i * (1 - x)/(1 - 2*x + x^(i+1))).

Sum_{k>0} T(n,k)*k = A336482(n).

A057892 Negabinary numbral addition table read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 2, 6, 2, 3, 3, 3, 3, 4, 0, 12, 0, 4, 5, 5, 13, 13, 5, 5, 6, 26, 6, 2, 6, 26, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 4, 0, 4, 24, 4, 0, 4, 8, 9, 9, 1, 1, 25, 25, 1, 1, 9, 9, 10, 14, 10, 6, 26, 30, 26, 6, 10, 14, 10, 11, 11, 11, 11, 27, 27, 27, 27, 11, 11, 11, 11, 12, 8, 52, 8, 12, 24, 4, 24

Offset: 0

Marc LeBrun, Sep 25 2000

Every negabinary numbral appears infinitely often (since every signed integer can be represented as a sum of two signed integers in infinitely many ways).

			a(4)=6 since a(4) corresponds to the table entry for [1]+[1]=1+1=2=4-2=[6].
a(24)=2 since a(24) corresponds to the table entry for [3]+[3]=(-1)+(-1)=-2=[2]. - _Sean A. Irvine_, Jul 11 2022

A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Sean A. Irvine, Java program (github)

Cf. A005351, A005352, A057893, A057894.

a(24) and a(84) corrected and title clarified by Sean A. Irvine, Jul 11 2022

A171682 Number of compositions of n with the smallest part in the first position.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 37, 72, 140, 275, 540, 1069, 2118, 4206, 8365, 16659, 33204, 66231, 132179, 263913, 527119, 1053113, 2104428, 4205987, 8407382, 16807410, 33603024, 67187111, 134343790, 268638648, 537198557, 1074270342, 2148336463, 4296343787, 8592156886, 17183457812, 34365534564

Offset: 1

Vladeta Jovovic, Dec 15 2009

First differences of A097939.

			The a(6)=20 such compositions of 6 are
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 1 3 ]
[ 5]  [ 1 1 2 1 1 ]
[ 6]  [ 1 1 2 2 ]
[ 7]  [ 1 1 3 1 ]
[ 8]  [ 1 1 4 ]
[ 9]  [ 1 2 1 1 1 ]
[10]  [ 1 2 1 2 ]
[11]  [ 1 2 2 1 ]
[12]  [ 1 2 3 ]
[13]  [ 1 3 1 1 ]
[14]  [ 1 3 2 ]
[15]  [ 1 4 1 ]
[16]  [ 1 5 ]
[17]  [ 2 2 2 ]
[18]  [ 2 4 ]
[19]  [ 3 3 ]
[20]  [ 6 ]
- _Joerg Arndt_, Jan 01 2013.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A079500.

nn=37;Drop[CoefficientList[Series[Sum[x^i/(1-x^i/(1-x)),{i,1,nn}],{x,0,nn}],x],1]  (* Geoffrey Critzer, Mar 12 2013 *)

N=66; x='x+O('x^N);
gf= (1-x) * sum(k=1,N, x^k/(1-x-x^k) );
Vec(gf)
/* Joerg Arndt, Jan 01 2013 */

G.f.: (1-x) * Sum_{k>=1} x^k/(1-x-x^k). [Joerg Arndt, Jan 01 2013]
a(n) ~ 2^(n-2). - Vaclav Kotesovec, Sep 10 2014
G.f.: Sum_{n>=1} q^n/(1-Sum_{k>=n} q^k). - Joerg Arndt, Jan 03 2024

Added more terms, Joerg Arndt, Jan 01 2013

A306269 Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 1, 0, 1, 5, 6, 5, 3, 2, 1, 1, 1, 0, 1, 5, 9, 7, 5, 3, 2, 1, 1, 1, 0, 1, 7, 12, 12, 8, 5, 3, 2, 1, 1, 1, 0, 1, 8, 17, 17, 13, 8, 5, 3, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 01 2019

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  2  1  1  1
  0  1  2  2  1  1  1
  0  1  3  3  2  1  1  1
  0  1  3  4  3  2  1  1  1
  0  1  5  6  5  3  2  1  1  1
  0  1  5  9  7  5  3  2  1  1  1
  0  1  7 12 12  8  5  3  2  1  1  1
  0  1  8 17 17 13  8  5  3  2  1  1  1
  0  1 10 25 26 20 14  8  5  3  2  1  1  1
  0  1 12 34 39 31 21 14  8  5  3  2  1  1  1
The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
  ((ooooooo))   (((oooooo)))    ((((ooooo))))    (((((oooo)))))
  ((o)(ooooo))  (((o)(oooo)))   ((((o)(ooo))))   (((((o)(oo)))))
  ((oo)(oooo))  (((oo)(ooo)))   ((((o))((oo))))
                (((o))((ooo)))

Alois P. Heinz, Rows n = 1..200, flattened
Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.

Row sums give A306201.
T(2n-1,n) gives A306274.
Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.

ubk[n_,k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#,k-1]&/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}],SameQ[k,##]&@@Length/@Position[#,{}]&];
Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]

A144406 Rectangular array A read by upward antidiagonals: entry A(n,k) in row n and column k gives the number of compositions of k in which no part exceeds n, n>=1, k>=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 5, 1, 1, 1, 2, 4, 7, 8, 1, 1, 1, 2, 4, 8, 13, 13, 1, 1, 1, 2, 4, 8, 15, 24, 21, 1, 1, 1, 2, 4, 8, 16, 29, 44, 34, 1, 1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1, 1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1, 1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 29 2008

Polynomial expansion as antidiagonal of p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)). Based on the Pisot general polynomial type q(x,n) = x^n - (x^n-1)/(x-1) (the original name of the sequence).
Row sums are 1, 2, 3, 5, 8, 14, ... (A079500).
Conjecture: Since the array row sequences successively tend to A000079, the absolute values of nonzero differences between two successive row sequences tend to A045623 = {1,2,5,12,28,64,144,320,704,1536,...}, as k -> infinity. - L. Edson Jeffery, Dec 26 2013

			Array A begins:
  {1, 1, 1, 1, 1,  1,  1,  1,   1,   1,   1, ...}
  {1, 1, 2, 3, 5,  8, 13, 21,  34,  55,  89, ...}
  {1, 1, 2, 4, 7, 13, 24, 44,  81, 149, 274, ...}
  {1, 1, 2, 4, 8, 15, 29, 56, 108, 208, 401, ...}
  {1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, ...}
  {1, 1, 2, 4, 8, 16, 32, 63, 125, 248, 492, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 504, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, ...}
  {1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, ...}
  ... - _L. Edson Jeffery_, Dec 26 2013
As a triangle:
  {1},
  {1, 1},
  {1, 1, 1},
  {1, 1, 2, 1},
  {1, 1, 2, 3, 1},
  {1, 1, 2, 4, 5, 1},
  {1, 1, 2, 4, 7, 8, 1},
  {1, 1, 2, 4, 8, 13, 13, 1},
  {1, 1, 2, 4, 8, 15, 24, 21, 1},
  {1, 1, 2, 4, 8, 16, 29, 44, 34, 1},
  {1, 1, 2, 4, 8, 16, 31, 56, 81, 55, 1},
  {1, 1, 2, 4, 8, 16, 32, 61, 108, 149, 89, 1},
  {1, 1, 2, 4, 8, 16, 32, 63, 120, 208, 274, 144, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 125, 236, 401, 504, 233, 1},
  {1, 1, 2, 4, 8, 16, 32, 64, 127, 248, 464, 773, 927, 377, 1}

Cf. A000079, A045623, A092921, A175331.
Same as A048887 but with a column of 1's added on the left (the number of compositions of 0 is defined to be equal to 1).
Array rows (with appropriate offsets) are A000012, A000045, A000073, A000078, A001591, A001592, etc.

g[x_, n_] = x^(n) - (x^n - 1)/(x - 1);
h[x_, n_] = FullSimplify[ExpandAll[x^(n)*g[1/x, n]]];
f[t_, n_] := 1/h[t, n];
a = Table[CoefficientList[Series[f[t, m], {t, 0, 30}], t], {m, 1, 31}];
b = Table[Table[a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}];
Flatten[b] (* Triangle version *)
Grid[Table[CoefficientList[Series[(1 - x)/(1 - 2 x + x^(n + 1)), {x, 0, 10}], x], {n, 1, 10}]] (* Array version - L. Edson Jeffery, Jul 18 2014 *)

t(n,m) = antidiagonal_expansion of p(x,n) where p(x,n) = (x-1)/(x^n*(-x+(2*x-1)/x^n)).

G.f. for array A: (1-x)/(1 - 2*x + x^(n+1)), n>=1. - L. Edson Jeffery, Dec 26 2013

Definition changed by L. Edson Jeffery, Jul 18 2014

cp's OEIS Frontend

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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A186537 G.f.: Sum( x^k/(1-2*x+x^k), k=1..oo).

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A320270 Number of unlabeled balanced semi-binary rooted trees with n nodes.

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A096202 Number of coverings of {1...n} by translation of a single set.

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A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A382312 Irregular triangle read by rows: T(n,k) is the number of compositions of n with k records.

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A057892 Negabinary numbral addition table read by antidiagonals.

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A171682 Number of compositions of n with the smallest part in the first position.

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