A007059 -id:A007059 - cp's OEIS frontend

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

0, 2, 10, 12, 42, 52, 56, 170, 204, 212, 232, 240, 682, 820, 844, 852, 920, 936, 976, 992, 2730, 3276, 3284, 3380, 3404, 3412, 3640, 3688, 3736, 3752, 3888, 3920, 4000, 4032, 10922, 13108, 13132, 13140, 13516, 13524, 13620, 13644, 13652, 14568, 14744, 14760

Offset: 1

Gus Wiseman, Nov 21 2022

nonn

An ordered tree is balanced if all leaves are the same distance from the root.

The binary encoding of an ordered tree (see A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

			The terms together with their corresponding trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  204: ((o)(o))
  212: ((ooo))
  232: (((oo)))
  240: ((((o))))
  682: (ooooo)
  820: ((o)(oo))
  844: ((oo)(o))
  852: ((oooo))
  920: (((o)(o)))
  936: (((ooo)))
  976: ((((oo))))
  992: (((((o)))))

These trees are counted by A007059.
This is a subset of A014486.
The version for binary trees is A057122.
The unordered version is A184155, counted by A048816.
Another ranking of balanced ordered trees is A358459.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A001263, A003238, A244925, A358377, A358379, A358523.

binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
Select[Range[0,1000],binbalQ[#]&&SameQ@@Length/@Position[bint[#],{}]&]

A066351 Bisection of A007059.

Original entry on oeis.org

1, 2, 5, 14, 43, 140, 472, 1628, 5719, 20388, 73562, 268066, 984911, 3643570, 13557020, 50691978, 190353370, 717457656, 2713061899, 10289600164, 39127877886, 149147692734, 569767908076, 2180978471298, 8363866011929, 32129445138352, 123618810558184

Offset: 0

N. J. A. Sloane, Dec 19 2001

R. Kemp, Balanced ordered trees, Random Structures and Algorithms, 5, 1994, 99-121.

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

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(2*n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, May 13 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_] := b[2*n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

Conjecture: a(n) ~ 0.721... * 4^n / n. - Vaclav Kotesovec, Aug 25 2014

More terms from Emeric Deutsch, Jun 10 2004

A039671 Row sums up to the main diagonal of the "postage stamp" array (n,m >= 0) defined in A007059.

Original entry on oeis.org

1, 1, 3, 8, 21, 53, 130, 310, 724, 1661, 3757, 8398, 18588, 40800, 88918, 192592, 414907, 889631, 1899554, 4040864, 8567342, 18109698, 38176280, 80278798, 168432854, 352658013, 736977583, 1537420460, 3202035086, 6658948608

Offset: 0

Richard L. Ollerton and Eirwyn L. Ollerton

			a(0)=1, a(1)=0+1=1, a(2)=0+1+2=3, a(3)=0+1+3+4=8.

Cf. A007059.

f[ n_, m_ ] := f[ n, m ]=Which[ n>0, Sum[ f[ n-i, m ], {i, 1, m} ], n<0, 0, n==0, 1 ] Table[ Sum[ f[ n, j ], {j, 0, n} ], {n, 0, 30} ]

Conjecture: a(n) = Sum_{k=1..n} A126198(n,k), for n > 0. - L. Edson Jeffery, Nov 29 2013

A066350 Bisection of A007059.

Original entry on oeis.org

0, 1, 3, 8, 24, 77, 256, 874, 3045, 10780, 38674, 140268, 513350, 1892875, 7023562, 26200182, 98182666, 369393466, 1394632365, 5281753216, 20058841004, 76370762838, 291436246116, 1114474774102, 4270050059100, 16389518073023, 63010289558056, 242614318886286

Offset: 0

N. J. A. Sloane, Dec 19 2001

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

Cf. A007059, A066351.

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(2*n-1, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 19 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_] := b[2n-1, 0];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

More terms from Joshua Zucker, May 12 2006

A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563

Offset: 1

Gus Wiseman, Nov 19 2022

nonn

An ordered tree is balanced if all leaves have the same distance from the root.

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))

Gus Wiseman, Statistics, classes, and transformations of standard compositions

These trees are counted by A007059.
The unordered version is A184155, counted by A048816.
A000108 counts ordered rooted trees, unordered A000081.
A358379 gives depth of standard ordered trees.
Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],SameQ@@Length/@Position[srt[#],{}]&]

A038495 Duplicate of A007059.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 14, 24, 43, 77, 140, 256, 472, 874, 1628, 3045, 5719, 10780, 20388

Offset: 0

dead

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

Original entry on oeis.org

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

A120803 Number of series-reduced balanced trees with n leaves.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 16, 20, 37, 47, 80, 111, 183, 256, 413, 591, 940, 1373, 2159, 3214, 5067, 7649, 12054, 18488, 29203, 45237, 71566, 111658, 176710, 276870, 437820, 687354, 1085577, 1705080, 2688285, 4221333, 6644088, 10425748

Offset: 1

Franklin T. Adams-Watters, Aug 18 2006

nonn

In other words, rooted trees with all leaves at the same level and no node having exactly one child; the order of children is not significant.

			From _Gus Wiseman_, Oct 07 2018: (Start)
The a(10) = 16 series-reduced balanced rooted trees:
  (oooooooooo)
  ((ooooo)(ooooo))
  ((oooo)(oooooo))
  ((ooo)(ooooooo))
  ((oo)(oooooooo))
  ((ooo)(ooo)(oooo))
  ((oo)(oooo)(oooo))
  ((oo)(ooo)(ooooo))
  ((oo)(oo)(oooooo))
  ((oo)(oo)(ooo)(ooo))
  ((oo)(oo)(oo)(oooo))
  ((oo)(oo)(oo)(oo)(oo))
  (((oo)(ooo))((oo)(ooo)))
  (((oo)(oo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oooo)))
  (((oo)(oo))((oo)(oo)(oo)))
(End)

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

Cf. A000081, A000669, A001003, A001678, A007059, A048816, A079500, A119262, A244925, A316624, A320154, A320160, A320169, A320179.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(u=vector(n), v=vector(n)); u[1]=1; while(u, v+=u; u=EulerT(u)-u); v} \\ Andrew Howroyd, Oct 26 2018

Let s_0(n) = 1 if n = 1, 0 otherwise; s_{k+1}(n) = EULER(s_k)(n) - s_k(n), where EULER is the Euler transform. Then a_n = sum_k s_k(n). (s_k(n) is the number of such trees of height k.) Note that s_k(n) = 0 for n < 2^k.

A048888 a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 42, 76, 139, 255, 471, 873, 1627, 3044, 5718, 10779, 20387, 38673, 73561, 140267, 268065, 513349, 984910, 1892874, 3643569, 7023561, 13557019, 26200181, 50691977, 98182665, 190353369, 369393465, 717457655

Offset: 0

Clark Kimberling

nonn

From Marc LeBrun, Dec 12 2001: (Start)
Define a "numbral arithmetic" by replacing addition with binary bitwise inclusive-OR (so that [3] + [5] = [7] etc.) and multiplication becomes shift-&-OR instead of shift-&-add (so that [3] * [3] = [7] etc.). [d] divides [n] means there exists an [e] with [d] * [e] = [n]. For example the six divisors of [14] are [1], [2], [3], [6], [7] and [14]. Then it appears that this sequence gives the number of proper divisors of [2^n-1]. Conjecture confirmed by Richard C. Schroeppel, Dec 14 2001. (End)
The number of "prime endofunctions" on n points, meaning the cardinality of the subset of the A001372(n) mappings (or mapping patterns) up to isomorphism from n (unlabeled) points to themselves (endofunctions) which are neither the sum of prime endofunctions (i.e., whose disjoint connected components are prime endofunctions) nor the categorical product of prime endofunctions. The n for which a(n) is prime (n such that the number of prime endofunctions on n points is itself prime) are 2, 4, 5, 6, 9, 13, 19, ... - Jonathan Vos Post, Nov 19 2006
For n>=1, compositions p(1)+p(2)+...+p(m)=n such that p(k)<=p(1)+1, see example. - Joerg Arndt, Dec 28 2012

			From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(6)=23 compositions p(1)+p(2)+...+p(m)=6 such that p(k)<=p(1)+1:
[ 1]  [ 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 2 ]
[ 3]  [ 1 1 1 2 1 ]
[ 4]  [ 1 1 2 1 1 ]
[ 5]  [ 1 1 2 2 ]
[ 6]  [ 1 2 1 1 1 ]
[ 7]  [ 1 2 1 2 ]
[ 8]  [ 1 2 2 1 ]
[ 9]  [ 2 1 1 1 1 ]
[10]  [ 2 1 1 2 ]
[11]  [ 2 1 2 1 ]
[12]  [ 2 1 3 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 3 1 1 1 ]
[17]  [ 3 1 2 ]
[18]  [ 3 2 1 ]
[19]  [ 3 3 ]
[20]  [ 4 1 1 ]
[21]  [ 4 2 ]
[22]  [ 5 1 ]
[23]  [ 6 ]
(End)

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.

Cf. A007059.

Cf. A000312, A001372, A002861, A006961, A001373, A054050, A054745, A125024.

N = 66;  x = 'x + O('x^N);
gf = sum(n=0,N,  (1-x^n)*x^n/(1-2*x+x^(n+1)) ) + 'c0;
v = Vec(gf);  v[1]-='c0;  v
/* Joerg Arndt, Apr 14 2013 */

G.f.: Sum_{k>0} x^k*(1-x^k)/(1-2*x+x^(k+1)). - Vladeta Jovovic, Feb 25 2003

a(m) = Sum_{ n=2..m+1 } Fn(m) where Fn is a Fibonacci n-step number (Fibonacci, tetranacci, etc.) indexed as in A000045, A000073, A000078. - Gerald McGarvey, Sep 25 2004

A067138 OR-numbral multiplication table, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 7, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 14, 20, 20, 14, 14, 8, 0, 0, 9, 16, 15, 24, 21, 24, 15, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 31, 28

Offset: 0

Jens Voß, Jan 02 2002

See A048888 for the definition of OR-numbral arithmetic

			The top left 0..16 x 0..16 corner of the array:
  0,  0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0,   0,
  0,  1,  2,  3,  4,  5,  6,   7,   8,   9,  10,  11,  12,  13,  14,  15,
  0,  2,  4,  6,  8, 10, 12,  14,  16,  18,  20,  22,  24,  26,  28,  30,
  0,  3,  6,  7, 12, 15, 14,  15,  24,  27,  30,  31,  28,  31,  30,  31,
  0,  4,  8, 12, 16, 20, 24,  28,  32,  36,  40,  44,  48,  52,  56,  60,
  0,  5, 10, 15, 20, 21, 30,  31,  40,  45,  42,  47,  60,  61,  62,  63,
  0,  6, 12, 14, 24, 30, 28,  30,  48,  54,  60,  62,  56,  62,  60,  62,
  0,  7, 14, 15, 28, 31, 30,  31,  56,  63,  62,  63,  60,  63,  62,  63,
  0,  8, 16, 24, 32, 40, 48,  56,  64,  72,  80,  88,  96, 104, 112, 120,
  0,  9, 18, 27, 36, 45, 54,  63,  72,  73,  90,  91, 108, 109, 126, 127,
  0, 10, 20, 30, 40, 42, 60,  62,  80,  90,  84,  94, 120, 122, 124, 126,
  0, 11, 22, 31, 44, 47, 62,  63,  88,  91,  94,  95, 124, 127, 126, 127,
  0, 12, 24, 28, 48, 60, 56,  60,  96, 108, 120, 124, 112, 124, 120, 124,
  0, 13, 26, 31, 52, 61, 62,  63, 104, 109, 122, 127, 124, 125, 126, 127,
  0, 14, 28, 30, 56, 62, 60,  62, 112, 126, 124, 126, 120, 126, 124, 126,
  0, 15, 30, 31, 60, 63, 62,  63, 120, 127, 126, 127, 124, 127, 126, 127,
  0, 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240,
.
Multiplying 3 ("11" in binary) with itself in this system means taking bitwise-or of "11" with itself, when shifted one bit-position left:
       11
      110
  -------
OR:   111 = 7 in decimal = A(3,3).
.
Multiplying 10 (= "1010" in binary) and 11 (= "1011" in binary) in this system means taking bitwise-or of binary number 1011 when shifted once left with the same binary number when shifted three bit-positions left:
      10110
    1011000
    -------
OR: 1011110 = 94 in decimal = A(10,11) = A(11,10).

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of square array

Cf. A003986, A067139, A048888, A007059, A067398 (main diagonal).

Cf. also A004247, A048720 for analogous multiplication tables.

t(n, k) = {res = 0; for (i=0, length(binary(n))-1, if (bittest(n, i), res = bitor(res, shift(k, i)));); return (res);} \\ Michel Marcus, Apr 14 2013

From Rémy Sigrist, Mar 17 2021: (Start)
T(n, 0) = 0.
T(n, 1) = n.
T(n, 2^k) = n*2^k for any k >= 0.
T(n, n) = A067398(n).
(End)
For all n, k: A048720(n,k) <= A(n,k) <= A004247(n,k). - Antti Karttunen, Mar 17 2021

Example-section rewritten by Antti Karttunen, Mar 17 2021

cp's OEIS Frontend

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

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A066351 Bisection of A007059.

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A039671 Row sums up to the main diagonal of the "postage stamp" array (n,m >= 0) defined in A007059.

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A066350 Bisection of A007059.

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A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

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A038495 Duplicate of A007059.

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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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A120803 Number of series-reduced balanced trees with n leaves.

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A048888 a(n) = Sum_{m=1..n} T(m,n+1-m), array T as in A048887.