A079500 -id:A079500 - cp's OEIS frontend

A316624 Number of balanced p-trees with n leaves.

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 40, 47, 83, 111, 201, 259, 454, 603, 1049, 1432, 2407, 3390, 6006, 8222, 13904, 20304, 34828, 50291, 85817, 126013, 217653, 317894, 535103, 798184, 1367585, 2008125, 3360067, 5048274, 8499942, 12623978, 21023718, 31552560, 52575257

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A p-tree of weight n is either a single node (if n = 1) or a finite sequence of p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(7) = 4 balanced p-trees:
  o  (oo)  (ooo)  (oooo)      (ooooo)      (oooooo)        (ooooooo)
                  ((oo)(oo))  ((ooo)(oo))  ((ooo)(ooo))    ((oooo)(ooo))
                                           ((oooo)(oo))    ((ooooo)(oo))
                                           ((oo)(oo)(oo))  ((ooo)(oo)(oo))

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

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

ptrs[n_]:=If[n==1,{"o"},Join@@Table[Tuples[ptrs/@p],{p,Rest[IntegerPartitions[n]]}]];
Table[Length[ptrs[n]],{n,12}]
Table[Length[Select[ptrs[n],SameQ@@Length/@Position[#,"o"]&]],{n,12}]

seq(n)={my(p=x + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(17) and beyond from Andrew Howroyd, Oct 26 2018

A067399 Number of divisors of n in OR-numbral arithmetic.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 3, 4, 2, 4, 2, 6, 2, 6, 5, 5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8, 6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14, 7, 2, 4, 2, 6, 2, 4, 2, 8, 3, 4, 2, 6, 2, 4, 2, 10, 2, 4, 2, 9, 5, 4, 2, 8, 2, 8, 4, 6, 2, 8, 6, 12, 2, 4, 4, 6

Offset: 1

Jens Voß, Jan 23 2002

nonn

See A048888 for the definition of OR-numbral arithmetic. The example shows that this sequence is not multiplicative.

In other words, number of lunar divisors of n in base 2.

			a(15)=5 since [15] has the 5 OR-numbral divisors [1], [3], [5], [7] and [15].
If written as a triangle with rows of lengths 1,2,4,8,16,...:
1,
2, 2,
3, 2, 4, 3,
4, 2, 4, 2, 6, 2, 6, 5,
5, 2, 4, 2, 6, 3, 4, 2, 8, 2, 4, 4, 9, 2, 10, 8,
6, 2, 4, 2, 6, 2, 4, 2, 8, 2, 6, 2, 6, 4, 4, 4, 10, 2, 4, 4, 6, 2, 8, 4, 12, 2, 4, 4, 15, 4, 16, 14,
...,
the last terms in each row give A079500(n). The penultimate terms in the rows give 2*A079500(n-1). - _N. J. A. Sloane_, Mar 05 2011

N. J. A. Sloane, Table of n, a(n) for n = 1..1024
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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

A079500 is the subsequence a(2^k-1). - N. J. A. Sloane, Feb 23 2011
Cf. A003986, A007059, A048888, A067138, A067139, A067398, A067400, A067401.
See A188548 for the sum of the divisors.

A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 12, 16, 25, 35, 53, 77, 117, 173, 265, 396, 605, 919, 1408, 2147, 3305, 5070, 7819, 12049, 18635, 28811, 44672, 69264, 107618, 167292, 260446, 405686, 632743, 987441, 1542555, 2411208, 3772247, 5905002, 9250436, 14499234, 22740910, 35686092

Offset: 0

Gus Wiseman, Jan 29 2019

nonn

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. The only balanced identity trees are rooted paths.

			The a(1) = 1 through a(7) = 8 balanced rooted semi-identity trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((((o))))  ((o)(oo))    ((o)(ooo))
                                     ((((oo))))   ((((ooo))))
                                     (((((o)))))  (((o)(oo)))
                                                  (((((oo)))))
                                                  ((((((o))))))

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000081, A004111, A048816, A079500, A120803, A306200, A316624, A317712, A320169, A320270.

Row sums of A306269.

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

More terms from Alois P. Heinz, Jan 29 2019

A067398 Squares in OR-numbral arithmetic.

Original entry on oeis.org

0, 1, 4, 7, 16, 21, 28, 31, 64, 73, 84, 95, 112, 125, 124, 127, 256, 273, 292, 311, 336, 341, 380, 383, 448, 473, 500, 511, 496, 509, 508, 511, 1024, 1057, 1092, 1127, 1168, 1205, 1244, 1279, 1344, 1385, 1364, 1407, 1520, 1533, 1532, 1535, 1792, 1841, 1892

Offset: 0

Jens Voß, Jan 23 2002

nonn

See A048888 for the definition of OR-numbral arithmetic.
Or, squares in lunar arithmetic base 2, written in base 10. - N. J. A. Sloane, Oct 02 2010
This sequence is not multiplicative; for example a(15) = 127 != 7 * 21 = a(3) * a(5). It is totally OR-numbral multiplicative: a([n] * [m]) = [a(n)] * [a(m)] in OR-numbral arithmetic. - Franklin T. Adams-Watters, Oct 27 2006

			A067398(5) = 21 since [5] * [5] = [21] in OR-numbral arithmetic.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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]
D. Applegate, M. LeBrun, N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
A. Frosini and S. Rinaldi, On the Sequence A079500 and Its Combinatorial Interpretations, J. Integer Seq., Vol. 9 (2006), Article 06.3.1.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A003986, A007059, A048888, A067138, A067139, A067399, A067400, A067401.

a067398 :: Integer -> Integer
a067398 0 = 0
a067398 n = orm n n where
   orm 1 v = v
   orm u v = orm (shiftR u 1) (shiftL v 1) .|. if odd u then v else 0
-- Reinhard Zumkeller, Mar 01 2013

A320169 Number of balanced enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 3, 6, 9, 20, 31, 70, 114, 243, 415, 961, 1603, 3564, 6559, 14913, 26630, 60037, 110160, 248859, 458445, 1001190, 1882350, 4220358, 7765303, 16822107, 32307240, 70081784, 133716083, 291788153, 561823990, 1230204229, 2396185727, 5176454708, 10220127290

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

An enriched p-tree of weight n is either the number n itself or a finite sequence of enriched p-trees whose weights are weakly decreasing and sum to n.

A tree is balanced if all leaves have the same height.

			The a(1) = 1 through a(6) = 20 balanced enriched p-trees:
  1  2     3      4           5            6
     (11)  (21)   (22)        (32)         (33)
           (111)  (31)        (41)         (42)
                  (211)       (221)        (51)
                  (1111)      (311)        (222)
                  ((11)(11))  (2111)       (321)
                              (11111)      (411)
                              ((21)(11))   (2211)
                              ((111)(11))  (3111)
                                           (21111)
                                           (111111)
                                           ((21)(21))
                                           ((22)(11))
                                           ((31)(11))
                                           ((111)(21))
                                           ((21)(111))
                                           ((211)(11))
                                           ((111)(111))
                                           ((1111)(11))
                                           ((11)(11)(11))

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

Cf. A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A196545, A289501, A319312.

Cf. A316624, A320154, A320155, A320160, A320179.

eptrs[n_]:=Prepend[Join@@Table[Tuples[eptrs/@p],{p,Rest[IntegerPartitions[n]]}],n];
Table[Length[Select[eptrs[n],SameQ@@Length/@Position[#,_Integer]&]],{n,12}]

seq(n)={my(p=x/(1-x) + O(x*x^n), q=0); while(p, q+=p; p = 1/prod(k=1, n, 1 - polcoef(p,k)*x^k + O(x*x^n)) - 1 - p); Vec(q)} \\ Andrew Howroyd, Oct 26 2018

Terms a(16) and beyond from Andrew Howroyd, Oct 26 2018

A320179 Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 6, 1, 0, 0, 0, 0, 0, 1, 7, 1, 0, 0, 0, 0, 0, 0, 1, 11, 4, 0, 0, 0, 0, 0, 0, 0, 1, 13, 6, 0, 0, 0, 0, 0, 0, 0, 0, 1, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 23, 0, 0, 0

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  0  0
  0  1  3  0  0  0
  0  1  3  0  0  0  0
  0  1  6  1  0  0  0  0
  0  1  7  1  0  0  0  0  0
  0  1 11  4  0  0  0  0  0  0
  0  1 13  6  0  0  0  0  0  0  0
  0  1 20 16  0  0  0  0  0  0  0  0
  0  1 23 23  0  0  0  0  0  0  0  0  0
  0  1 33 46  0  0  0  0  0  0  0  0  0  0
The T(10,3) = 4 rooted trees:
   (((oo)(oo))((oo)(oooo)))
   (((oo)(oo))((ooo)(ooo)))
   (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A120803. Third column is A083751. An irregular version is A320221.
Cf. A000669, A001678, A048816, A079500, A119262, A244925, A316655, A319312.
Cf. A316624, A320154, A320155, A320160, A320172, A320173.

qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,14},{k,0,n-1}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); vector(n, n, Vecrev(v[n], n))}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

A156041 Array A(n,k) (n>=1, k>=1) read by antidiagonals, where A(n,k) is the number of compositions (ordered partitions) of n into exactly k parts, some of which may be zero, with the first part greater than or equal to all the rest.

Original entry on oeis.org

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

Offset: 1

Jack W Grahl, Feb 02 2009, Feb 11 2009

A(n,k) is of course smaller than the number of ordered partitions of n into k parts and at least the number of partitions into k parts in descending order.
The sums of the antidiagonals give A079500 - 1. - N. J. A. Sloane, Feb 26 2011
For an alternative definition of essentially the same sequence, as a triangle, and which avoids the use of parts of size zero, see A184957. - N. J. A. Sloane, Feb 27 2011

			The array A(n,k) begins:
  1  1  1  1  1  1  1  1  1 ...
  1  2  3  4  5  6  7  8  9 ...
  1  2  4  7 11 16 22 29 ...
  1  3  6 11 19 31 48 ...
  1  3  8 17 32 56 ...
  1  4 11 26 54 ...
  1  4 13 35 ...
  ...
The antidiagonals are:
  1,
  1, 1,
  1, 2, 1,
  1, 2, 3, 1,
  1, 3, 4, 4, 1,
  1, 3, 6, 7, 5, 1,
  1, 4, 8, 11, 11, 6, 1,
  1, 4, 11, 17, 19, 16, 7, 1,
  1, 5, 13, 26, 32, 31, 22, 8, 1,
  ...
A(3,5) = 11 and the 11 partition of 3 into 5 parts of this type are: (3,0,0,0,0), (2,1,0,0,0), (2,0,1,0,0), (2,0,0,1,0), (2,0,0,0,1), (1,1,1,0,0), (1,1,0,1,0), (1,1,0,0,1), (1,0,1,1,0), (1,0,1,0,1), (1,0,0,1,1).

R. H. Hardin, Table of n, a(n) for n = 1..2278
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 32.

A156039 gives A(n,4) and A156040 gives A(n,3). A156042 is the part on or below the main diagonal. A(n,2) is A008619. A(2,n) is A000027. A(3,n) is A000124.

Cf. A079500.

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-1), d=1..14); # Alois P. Heinz, Jun 14 2009

(* Returns rectangular array *) 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,nn}],{n,1,nn}]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

A(n,k) = [x^n] Sum_{i=0..n} x^i*((1 - x^(i+1))/(1-x))^(k-1). - Geoffrey Critzer, Jul 15 2013

More terms from Alois P. Heinz, Jun 14 2009

Edited by N. J. A. Sloane, Feb 26 2011

A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

Original entry on oeis.org

1, 1, 1, 4, 11, 41, 162, 1030, 7205, 55522, 442443, 3810852, 35272030, 351697516, 3735838550, 42719792640, 529195988635, 7128835815387, 103651381499810, 1610812109555323, 26489497655582729, 457497408108551450, 8248899117402701046, 154624472715479106919

Offset: 1

Gus Wiseman, Oct 06 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.

			The a(1) = 1 through a(5) = 11 rooted trees:
  1  (12)  (123)    (1234)      (12345)
                  ((12)(34))  ((12)(345))
                  ((13)(24))  ((13)(245))
                  ((14)(23))  ((14)(235))
                              ((15)(234))
                              ((23)(145))
                              ((24)(135))
                              ((25)(134))
                              ((34)(125))
                              ((35)(124))
                              ((45)(123))

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A000311, A000669, A001678, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A320154, A320160, A316624, A320169, A320173, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
phy2[labs_]:=If[Length[labs]==1,labs,Union@@Table[Sort/@Tuples[phy2/@ptn],{ptn,Select[sps[Sort[labs]],Length[#1]>1&]}]];
Table[Length[Select[phy2[Range[n]],SameQ@@Length/@Position[#,_Integer]&]],{n,7}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(u=vector(n), v=vector(n)); u[1]=k; while(u, v+=u; u=EulerT(u)-u); v}
seq(n)={my(M=Mat(vectorv(n,k,b(n,k)))); vector(n, k, sum(i=1, k, binomial(k,i)*(-1)^(k-i)*M[i,k]))} \\ Andrew Howroyd, Oct 26 2018

E.g.f. A(x) satisfies A(x) = x + A(exp(x)-x-1). - Ira M. Gessel, Nov 17 2021

Terms a(10) and beyond from Andrew Howroyd, Oct 26 2018

A320173 Number of inequivalent colorings of series-reduced balanced rooted trees with n leaves.

Original entry on oeis.org

1, 2, 3, 12, 23, 84, 204, 830, 2940, 13397, 58794, 283132, 1377302, 7087164, 37654377, 209943842, 1226495407, 7579549767, 49541194089, 341964495985, 2476907459261, 18703210872343, 146284738788714, 1179199861398539, 9760466433602510, 82758834102114911, 717807201648148643

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.

			Inequivalent representatives of the a(1) = 1 through a(5) = 23 colorings:
  1  (11)  (111)    (1111)      (11111)
     (12)  (112)    (1112)      (11112)
           (123)    (1122)      (11122)
                    (1123)      (11123)
                    (1234)      (11223)
                  ((11)(11))    (11234)
                  ((11)(12))    (12345)
                  ((11)(22))  ((11)(111))
                  ((11)(23))  ((11)(112))
                  ((12)(12))  ((11)(122))
                  ((12)(13))  ((11)(123))
                  ((12)(34))  ((11)(223))
                              ((11)(234))
                              ((12)(111))
                              ((12)(112))
                              ((12)(113))
                              ((12)(123))
                              ((12)(134))
                              ((12)(345))
                              ((13)(122))
                              ((22)(111))
                              ((23)(111))
                              ((23)(114))

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A244925, A319312.

Cf. A320154, A320155, A320160, A320174, A320175, A320179, A339645.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=x*sv(1) + O(x*x^n), q=0); while(p, q+=p; p=sEulerT(p)-1-p); q}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 11 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 11 2020

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

cp's OEIS Frontend

A316624 Number of balanced p-trees with n leaves.

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A067399 Number of divisors of n in OR-numbral arithmetic.

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A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.

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A067398 Squares in OR-numbral arithmetic.

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Haskell

A320169 Number of balanced enriched p-trees of weight n.

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A320179 Regular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k.

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A156041 Array A(n,k) (n>=1, k>=1) read by antidiagonals, where A(n,k) is the number of compositions (ordered partitions) of n into exactly k parts, some of which may be zero, with the first part greater than or equal to all the rest.

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Maple

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A320155 Number of series-reduced balanced rooted trees with n labeled leaves.

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