A002572 -id:A002572 - cp's OEIS frontend

A049286 Triangle of partitions v(d,c) defined in A002572.

1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 5, 4, 2, 1, 1, 9, 7, 4, 2, 1, 1, 16, 12, 7, 4, 2, 1, 1, 28, 22, 13, 7, 4, 2, 1, 1, 50, 39, 24, 13, 7, 4, 2, 1, 1, 89, 70, 42, 24, 13, 7, 4, 2, 1, 1, 159, 126, 76, 43, 24, 13, 7, 4, 2, 1, 1, 285, 225, 137, 78, 43, 24, 13, 7, 4, 2, 1, 1, 510

Offset: 1

N. J. A. Sloane, Michael Somos

Rows are the columns in the table at the end of the Minc reference, read top to bottom. - Joerg Arndt, Jan 15 2024

			Triangle begins
    1,
    1,   1,
    2,   1,   1,
    3,   2,   1,  1,
    5,   4,   2,  1,  1,
    9,   7,   4,  2,  1,  1,
   16,  12,   7,  4,  2,  1,  1,
   28,  22,  13,  7,  4,  2,  1, 1,
   50,  39,  24, 13,  7,  4,  2, 1, 1,
   89,  70,  42, 24, 13,  7,  4, 2, 1, 1,
  159, 126,  76, 43, 24, 13,  7, 4, 2, 1, 1,
  285, 225, 137, 78, 43, 24, 13, 7, 4, 2, 1, 1,
  ...
Rows read backward approach A002843. - _Joerg Arndt_, Jan 15 2024

Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.

Cf. A002572, A002573, A002574, A049284, A049285.

See A047913 for another version.

v := proc(c,d) option remember; local i; if d < 0 or c < 0 then 0 elif d = c then 1 else add(v(i,d-c),i=1..2*c); fi; end;

v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; Table[v[d, c], {c, 1, 13}, {d, 1, c}] // Flatten (* Jean-François Alcover, Dec 10 2012, after Maple *)

A001180 Erroneous version of A002572.

Original entry on oeis.org

1, 1, 2, 3, 3, 5, 9, 16, 28, 50, 89, 159, 285, 510, 914, 1639, 2938, 5269, 9451, 16952

Offset: 1

dead

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A002846 Number of ways of transforming a set of n indistinguishable objects into n singletons via a sequence of n-1 refinements.

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 33, 116, 435, 1832, 8167, 39700, 201785, 1099449, 6237505, 37406458, 232176847, 1513796040, 10162373172, 71158660160, 511957012509, 3819416719742, 29195604706757, 230713267586731, 1861978821637735, 15484368121967620, 131388840051760458

Offset: 1

N. J. A. Sloane. Entry revised by N. J. A. Sloane, Jun 11 2012

Construct the ranked poset L(n) whose nodes are the A000041(n) partitions of n, with all the partitions into the same number of parts having the same rank. A partition into k parts is joined to a partition into k+1 parts if the latter is a refinement of the former.
The partition n^1 is at the left and the partition 1^n at the right. The illustration by Olivier Gérard shows the posets L(2) through L(8).
Then a(n) is the number of paths of length n-1 in L(n) that join n^1 to 1^n.
Stated another way, a(n) is the number of maximal chains in the ranked poset L(n). (This poset is not a lattice for n > 4.) - Comments corrected by Gus Wiseman, May 01 2016

			a(5) = 4 because there are 4 paths from top to bottom in this lattice:
  .
       ooooo
     /      \
  o.oooo   oo.ooo
    |    X    |
  o.o.ooo  o.oo.oo
     \       /
      o.o.o.oo
          |
      o.o.o.o.o
  .
(This is the ranked poset L(5), but drawn vertically rather than horizontally.)

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..80
P. Erdős, R. K. Guy and J. W. Moon, On refining partitions, J. London Math. Soc., 9 (1975), 565-570.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
Olivier Gérard, The ranked posets L(2),...,L(8)
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 1, [Cached copy, with permission]
Gus Wiseman, Hasse Diagrams of Partition Refinement Posets n=1..9, Version 2, [Cached copy, with permission]

See A213242, A213385, A213427 for related sequences, A327643.

v:= l-> [seq(`if`(i=1 or l[i]>l[i-1], seq(subs(1=[][], sort(subsop(
         i=[j, l[i]-j][], l))), j=1..l[i]/2), [][]), i=1..nops(l))]:
b:= proc(l) option remember; `if`(max(l)<2, 1, add(b(h), h=v(l))) end:
a:= n-> b([n]):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 22 2019

Mathematica
```
<Mitch Harris, Jan 19 2006 *)
```

def A002846(n): return Posets.IntegerPartitions(n).chain_polynomial().leading_coefficient()  # Max Alekseyev, Dec 23 2015

a(17)-a(25) from Mitch Harris, Jan 19 2006

A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 232, 1161, 6643, 44566, 327064, 2709050, 24312028, 240833770, 2546215687, 29251369570, 355838858402, 4658866773664, 64127566159756, 940320691236206

Offset: 0

Jonas Wallgren, Mar 17 2005

			{{{1,2,3},{4,5,6},{7,8,9}}, {{1,2,3},{4,6,8},{5,7,9}}, {{1,3,5},{2,4,6},{7,8,9}}, {{1,4,7},{2,5,8},{3,6,9}}, {{1,5,9},{2,3,4},{6,7,8}}} are the 5 ways to split 1, 2, 3, ..., 9 into 3 arithmetic progressions each with 3 elements. Thus a(3)=5.

R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971.
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics, in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]. See sequence "M".
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] Gives a(0)-a(10).

Cf. A104430-A104443.
All of A279197, A279198, A202705, A279199, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
See also A002848, A002849, A334250.

a(11)-a(14) from Alois P. Heinz, Dec 28 2011
a(15)-a(17) from Fausto A. C. Cariboni, Feb 22 2017
a(18)-a(19) from Martin Fuller, Jul 08 2025

A002843 Number of partitions of n into parts 1/2, 3/4, 7/8, 15/16, etc.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 43, 78, 141, 253, 456, 820, 1472, 2645, 4749, 8523, 15299, 27456, 49267, 88407, 158630, 284622, 510683, 916271, 1643963, 2949570, 5292027, 9494758, 17035112, 30563634, 54835835, 98383803, 176515310, 316694823, 568197628, 1019430782

Offset: 0

N. J. A. Sloane

Row sums of A049286 and A047913. [Vladeta Jovovic, Dec 02 2009]

Also number of compositions (a_1,a_2,...) of n with each a_i <= 2*a_(i-1). [Vladeta Jovovic, Dec 02 2009]

			A straightforward partition problem: 1 = 1/2 + 1/2 and there is no other partition of 1, so a(1)=1.
a(3)=4 since 3 = 6(1/2) = 4(3/4) = 2(3/4) + 3(1/2) = 2(7/8) + 3/4 + 1/2.
a(4)=7 since 4 = 8(1/2) = 5(1/2) + 2(3/4) = 2(1/2) + 4(3/4) = 3(1/2) + 3/4 + 2(7/8) = 3(3/4) + 2(7/8) = 1/2 + 4(7/8) = 2(15/16) + 7/8 + 3/4 + 1/2.
From _Joerg Arndt_, Dec 28 2012: (Start)
There are a(6)=24 compositions of 6 where part(k) <= 2 * part(k-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]  [ 1 2 3 ]
[10]  [ 2 1 1 1 1 ]
[11]  [ 2 1 1 2 ]
[12]  [ 2 1 2 1 ]
[13]  [ 2 2 1 1 ]
[14]  [ 2 2 2 ]
[15]  [ 2 3 1 ]
[16]  [ 2 4 ]
[17]  [ 3 1 1 1 ]
[18]  [ 3 1 2 ]
[19]  [ 3 2 1 ]
[20]  [ 3 3 ]
[21]  [ 4 1 1 ]
[22]  [ 4 2 ]
[23]  [ 5 1 ]
[24]  [ 6 ]
(End)

Minc, H.; A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid. Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 201 terms from Vincenzo Librandi)
David Benson, Pavel Etingof, On cohomology in symmetric tensor categories in prime characteristic, arXiv:2008.13149 [math.RT], 2020.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
H. Minc, A problem in partitions: Enumeration of elements of a given degree in the free commutative entropic cyclic groupoid, Proc. Edinburgh Math. Soc. (2) 11 1958/1959 223-224.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A047913, A049286.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, min(n-j, 2*j)), j=1..i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 24 2017

v[c_, d_] := v[c, d] = If[d < 0 || c < 0, 0, If[d == c, 1, Sum[v[i, d - c], {i, 1, 2*c}]]]; Join[{1}, Plus @@@ Table[v[d, c], {c, 1, 34}, {d, 1, c}]] (* Jean-François Alcover, Dec 10 2012, after Vladeta Jovovic *)

The g.f. (z**2+z+1)*(z-1)**2/(1-2*z-z**3+3*z**4) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

More terms from John W. Layman, Nov 24 2001
Examples and offset corrected by Larry Reeves (larryr(AT)acm.org), Jan 06 2005
Further terms from Vladeta Jovovic, Mar 13 2006

A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 36, 78, 171, 379, 851, 1928, 4396, 10087, 23273, 53948, 125608, 293543, 688366, 1619087, 3818818, 9029719, 21400706, 50828664, 120963298, 288405081, 688821573, 1647853491, 3948189131, 9473431479

Offset: 1

N. J. A. Sloane

a(n) <= A002955(n). - Max Alekseyev, Sep 23 2009

			From _M. F. Hasler_, Apr 17 2024: (Start)
The table with explicit lists of values starts as follows:
   n | distinct values of 2^...^2 with all possible parenthesizations
-----+---------------------------------------------------------------
   1 | 2
   2 | 2^2 = 4
   3 | (2^2)^2 = 2^(2^2) = 16
   4 | (2^2^2)^2 = 2^8 = 256, (2^2)^(2^2) = 2^(2^2^2) = 2^16 (= 65536)
   5 | 256^2 = 2^16, (2^16)^2 = 2^32, 2^256, 2^2^16 (~ 2*10^19728)
   6 | (2^16)^2 = 2^32, 2^64, 2^512, 2^2^16, 2^2^17, 2^2^32, 2^2^256, 2^2^2^16
   7 | 2^64, 2^128, 2^256, 2^1024, 2^2^17, 2^2^18, 2^2^32, 2^2^33, 2^2^64, 2^2^257,
     | 2^2^512, 2^2^2^16, 2^2^65537, 2^2^2^17, 2^2^2^32, 2^2^2^256, 2^2^2^2^16
  ...| ...
(When parentheses are omitted above, we use that ^ is right associative.) (End)

J. Q. Longyear, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

F. Goebel and R. P. Nederpelt, The number of numerical outcomes of iterated powers, Amer. Math. Monthly, 80 (1971), 1097-1103.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis, Amer. Math. Monthly 80 (8) (1973), 868-876.
R. K. Guy and J. L. Selfridge, The nesting and roosting habits of the laddered parenthesis (annotated cached copy, with permission)
Sean A. Irvine, Java program (github).
J. Longyear, T. Trotter, and N. J. A. Sloane, Correspondence.
MathOverflow, MathOverflow discussion of related questions.
Vladimir Reshetnikov, Computes terms of OEIS sequence A002845 (C# library).
Jon E. Schoenfield, The 851 values for n=12.
Index entries for sequences related to parenthesizing

Cf. A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A000081.

/* Define operators for numbers represented (recursively) as list of positions of bits 1. Illustration using the commands below: T = 3.bits; T.int */
n.bits = vector(hammingweight(n), v,  n -= 1 << v= valuation(n, 2); v.bits)
ONE = 1.bits; m.int = sum(i=1, #m, 1<=0])}
{ADD(m, n, a=#m, b=#n)= if(!a, n, !b, m, a=b=1; until(a>#m|| b>#n, if(m[a]==n[b], until(a>=#m|| m[a]!=m[a+1]|| !#m=m[^a], m[a]=ADD(m[a],ONE)); b++, CMP(m[a], n[b])<0, a++, m=concat([m[1..a-1], [n[b]], m[a..#m]]); b++)); b>#n|| m=concat(m,n[b..#n]); m)}
{CMP(m, n, a=#m, b=#n, c=0)= if(!b, a, !a, -1, while(!(c=CMP(m[a], n[b]))&& a--&& b--, ); if(c, c, 1-b))}
{SUB(m, n, a=#n)= if(!a, m, my(b=a=1, c, i); while(a<=#m && b<=#n, if(0>c=CMP(m[a], n[b]), a++, c, i=[c=n[b]]; b++; while(m[a]!=c=ADD(c, ONE), if(b<=#n && c==n[b], b++, i=concat(i, [c]))); m=concat([m[1..a-1], i, m[a+1..#m]]); a += #i, m=m[^a]; b++)); m)}
A2845 = List([[2.bits]]) /* List of values for each n */
{A002845(n)= while(#A2845= 15. - M. F. Hasler, Apr 28 2024

a(12)-a(13) corrected and a(14)-a(27) added by Jon E. Schoenfield, Oct 11 2008
a(28)-a(29) computed by Kirill Osenkov, added by Vladimir Reshetnikov, Feb 07 2019
a(30)-a(31) added by Sean A. Irvine, Feb 18 2019

A213385 a(n) = number of refinements of the partition n^1.

Original entry on oeis.org

1, 2, 3, 7, 15, 43, 131, 468, 1776, 7559, 34022, 166749, 853823, 4682358, 26720781, 161074458, 1004485751, 6576974188, 44322716809, 311440019349, 2247888977510, 16819336465164, 128915407382036, 1021269823516449, 8261243728564640, 68848043979970646

Offset: 1

N. J. A. Sloane, Jun 10 2012

nonn

Consider the ranked poset L(n) of partitions defined in A002846. Then a(n) is the total number of paths of all lengths 0,1,...,n-1 that start at n^1 and end at a node in the poset.

			Referring to the ranked poset L(5) shown in the example in A002846, there are 15 paths that start at ooooo:
end point / number of paths
ooooo / 1
o oooo / 1
oo ooo / 1
o o ooo / 2
o oo oo / 2
o o o oo / 4
o o o o o / 4
Total a(5) = 15.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..50
Olivier Gérard, The ranked posets L(2),...,L(8)
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence labeled H.

Cf. A002846, A213242, A213427.

b:= proc(l) option remember; local n, i, j, t; n:=nops(l);
      `if`(l[n]=1 and {l[1..n-1][]} minus {0}={}, 1,
      add(`if`(l[i]=0, 0, add(`if`(l[j]=0 or i=j and l[j]<2, 0,
      b([seq(`if`(t>n, 0, l[t])-`if`(t=i and t=j, 2, `if`(t=i or t=j,
      1, `if`(t=i+j, -1, 0))), t=1..max(n, i+j))])), j=i..n)), i=1..n))
    end:
g:= proc(n, i, l)
      `if`(n=0 and i=0, b(l), `if`(i=1, b([n, l[]]), add(g(n-i*j, i-1,
      `if`(l=[] and j=0, l, [j, l[]])), j=0..n/i)))
    end:
a:= n-> g(n, n, []):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 11 2012

b[l_List] := b[l] = Module[{n, i, j, t}, n = Length[l]; If[l[[n]] == 1 && Union[ l[[1 ;; n-1]]] ~Complement~ {0} == {}, 1, Sum[If[l[[i]] == 0, 0,  Sum[If[l[[j]] == 0 || i == j && l[[j]]<2, 0, b[Table[If[t>n, 0, l[[t]]] - Which[t == i && t == j, 2, t == i || t == j, 1, t == i+j, -1, True, 0], {t, 1, Max[n, i+j]}]]], {j, i, n}] ], {i, 1, n}]]]; g[n_, i_, l_List] := If[n == 0 && i == 0, b[l], If[i == 1, b[ Join[{n}, l]], Sum[g[n-i*j, i-1, If[l == {} && j == 0, l, Join[{j}, l]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

Definition clarified by David Applegate, Jun 10 2012
More terms from Alois P. Heinz, Jun 11 2012
Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A007178 Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.

Original entry on oeis.org

1, 1, 3, 13, 75, 525, 4347, 41245, 441675, 5259885, 68958747, 986533053, 15292855019, 255321427725, 4567457001915, 87156877087069, 1767115200924299, 37936303950503853, 859663073472084315, 20505904049009202685, 513593410566661282347, 13476082013068430626893

Offset: 1

N. J. A. Sloane, Simon Plouffe, Don Knuth

Also the dimension of the arity n component of the operad of level algebras (see the reference by Chataur-Livernet by definition), and the cardinality of the subset of the free commutative medial magma with n generators that contains each generator exactly once. The linear operad of level algebras is the linearization of the set operad of commutative medial magmas; the statement about commutative medial magmas follows from the description in the paper of Ježek-Kepka. - Vladimir Dotsenko, Mar 12 2022

			For n=3, the 3 sums are 1/2 + 1/4 + 1/4, 1/4 + 1/2 + 1/4, and 1/4 + 1/4 + 1/2.

D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..300
D. Chataur and M. Livernet, Adem-Cartan operads, arXiv:math/0209363 [math.AT], 2002-2003; Communications in Algebra 33 (2005), 4337-4360.
A. Giorgilli and G. Molteni, Representation of a 2-power as sum of k 2-powers: a recursive formula, J. Number Theory 133 (2013), no. 4, 1251-1261.
Jia Huang and Erkko Lehtonen, Associative-commutative spectra for some varieties of groupoids, arXiv:2401.15786 [math.CO], 2024. See p. 18.
J. Ježek and T. Kepka, Free entropic groupoids, Commentationes Mathematicae Universitatis Carolinae,Tome 022 (1981), p. 223-233.
D. E. Knuth, Letter to R. E. Tarjan & N. J. A. Sloane, Jul. 1975
Daniel Krenn and Stephan Wagner, Compositions into Powers of b : Asymptotic Enumeration and Parameters, arXiv:1410.4331 [math.NT], 2014.
S. Lehr, J. Shallit and J. Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
G. Molteni, Cancellation in a short exponential sum, J. Number Theory 130 (2010), no. 9, 2011-2027.
G. Molteni, Representation of a 2-power as sum of k 2-powers: the asymptotic behavior, Int. J. Number Theory 8 (2012), no. 8, 1923-1963.

Cf. A002572, A294746, A323840.

b:= proc(n, r, p) option remember; `if`(n b(n, 1, 0):
seq(a(n), n=1..23);  # Alois P. Heinz, Nov 07 2017

f[n_] := Coefficient[Expand[Sum[z^(2^j), {j, n}]^n], z, 2^n]; Array[f, 20] (* Robert G. Wilson v, Apr 08 2012 *)

f(n)={my(M);if(n>1,M=matrix(n,n);M[2,1] = 1;for(k=3,n,for(l=1,k-2,M[k,l] = 0;smx = min(2*l,k-l-1);for(s=1,smx, M[k,l] += binomial(k+l-1,2*l-s)*M[k-l,s]));M[k,k-1] = 1);M[n,1],1)}

a(n) = coefficient of z^(2^n) in (z+z^2+z^4+...+z^(2^n))^n. - Don Knuth.
From Giuseppe Molteni, Dec 14 2012: (Start)
Limit_{n->oo} (a(n)/n!)^(1/n) = 1.192674341213466032221288982528755... (see References: "Representation of a 2-power as sum of k 2-powers: the asymptotic behavior").
a(n) == 4 + (-1)^n (mod 8) for n > 2 (see References: "Representation of a 2-power as sum of k 2-powers: a recursive formula"). (End)
More precise asymptotics: a(n) ~ c * d^n * n!, where d = 1.192674341213466032221288982528755176734489232027131552652821007498903522051783..., c = 0.24849369086953813603231092781945750388624874631949260927875431616785914609... - Vaclav Kotesovec, Sep 20 2019
a(n) = A323840(n,n). - Alois P. Heinz, Mar 31 2021

More terms from Hugo van der Sanden

Minor edits from Vaclav Kotesovec, Jul 26 2014

A294775 Number A(n,k) of partitions of 1 into exactly k*n+1 powers of 1/(k+1); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 4, 5, 1, 1, 1, 1, 2, 4, 7, 9, 1, 1, 1, 1, 2, 4, 8, 13, 16, 1, 1, 1, 1, 2, 4, 8, 15, 25, 28, 1, 1, 1, 1, 2, 4, 8, 16, 29, 48, 50, 1, 1, 1, 1, 2, 4, 8, 16, 31, 57, 92, 89, 1, 1, 1, 1, 2, 4, 8, 16, 32, 61, 112, 176, 159, 1

Offset: 0

Alois P. Heinz, Nov 08 2017

			A(4,1) = 3: [1/4,1/4,1/4,1/8,1/8], [1/2,1/8,1/8,1/8,1/8], [1/2,1/4,1/8,1/16,1/16].
A(5,2) = 7: [1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/9,1/9,1/9,1/9,1/9,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27,1/27], [1/3,1/3,1/9,1/27,1/27,1/27,1/27,1/27,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/81,1/81,1/81,1/81,1/81,1/81], [1/3,1/3,1/9,1/9,1/27,1/27,1/81,1/81,1/243,1/243,1/243].
Square array A(n,k) begins:
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
  1,  2,  2,  2,  2,  2,  2,  2,  2, ...
  1,  3,  4,  4,  4,  4,  4,  4,  4, ...
  1,  5,  7,  8,  8,  8,  8,  8,  8, ...
  1,  9, 13, 15, 16, 16, 16, 16, 16, ...
  1, 16, 25, 29, 31, 32, 32, 32, 32, ...
  1, 28, 48, 57, 61, 63, 64, 64, 64, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Christian Elsholtz, Clemens Heuberger, Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.

Columns k=0-10 give (offsets may differ): A000012, A002572, A176485, A176503, A194628, A194629, A194630, A194631, A194632, A194633, A295081.
Main diagonal gives A011782(n-1) for n>0.
Cf. A294746.

b:= proc(n, r, k) option remember;
      `if`(n `if`(k=0, 1, b(k*n+1, 1, k+1)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, r_, k_] := b[n, r, k] = If[n < r, 0, If[r == 0, If[n == 0, 1, 0], Sum[b[n - j, k*(r - j), k], {j, 0, Min[n, r]}]]];
A[n_, k_] := If[k == 0, 1, b[k*n + 1, 1, k + 1]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Alois P. Heinz *)

A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 3, 10, 25, 12, 42, 8, 40, 204, 21, 135, 1002, 4228, 720, 5134, 29546, 4079, 35533, 3040, 28777, 281504, 20505, 212283, 2352469, 16907265, 1669221, 19424213, 167977344, 14708525, 191825926, 10567748, 149151774, 2102286756, 103372655, 1534969405

Offset: 1

N. J. A. Sloane

nonn

			For n = 3, the unique solution is 1 + 2 = 3.
For n = 12, there are 8 solutions:
  1  5  6 | 1  5  6 | 2  5  7 | 1  6  7
  2  8 10 | 3  7 10 | 3  6  9 | 4  5  9
  4  7 11 | 2  9 11 | 1 10 11 | 3  8 11
  3  9 12 | 4  8 12 | 4  8 12 | 2 10 12
  --------+---------+---------+--------
  2  4  6 | 2  6  8 | 3  4  7 | 3  5  8
  1  9 10 | 4  5  9 | 1  8  9 | 2  7  9
  3  8 11 | 3  7 10 | 5  6 11 | 4  6 10
  5  7 12 | 1 11 12 | 2 10 12 | 1 11 12

R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, "Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics," in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Martin Fuller, Table of n, a(n) for n = 1..52 (terms 1..42 from Fausto A. C. Cariboni, terms 43..44 from Frank Niedermeyer)
R. K. Guy, Letter to N. J. A. Sloane, Jun 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.
Matheplanet Calculating sequence element a(16) of OEIS A108235
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A002848, A108235, A161826.

nxyz(v,t)=local(n,r,x2); r=0; if(t==0,return(1)); for(i3=3*t,#v, n=v[i3]; for(i1=1,i3-2, x2=n-v[i1]; if(x2<=v[i1],break); for(i2=i1+1,i3-1, if(v[i2]>=x2, if(v[i2]==x2, r+=nxyz(vector(i3-3,k,v[if(kFranklin T. Adams-Watters

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others
a(32)-a(39) from Max Alekseyev, Feb 23 2012
Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023
a(40)-a(41) from Fausto A. C. Cariboni, Feb 04 2017
a(42) from Fausto A. C. Cariboni, Mar 12 2017

cp's OEIS Frontend

A049286 Triangle of partitions v(d,c) defined in A002572.

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A001180 Erroneous version of A002572.

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A002846 Number of ways of transforming a set of n indistinguishable objects into n singletons via a sequence of n-1 refinements.

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A104429 Number of ways to split {1, 2, 3, ..., 3n} into n arithmetic progressions each with 3 terms.

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A002843 Number of partitions of n into parts 1/2, 3/4, 7/8, 15/16, etc.

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A002845 Number of distinct values taken by 2^2^...^2 (with n 2's and parentheses inserted in all possible ways).

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A213385 a(n) = number of refinements of the partition n^1.

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A007178 Number of ways to write 1 as ordered sum of n powers of 1/2, allowing repeats.