A047913 -id:A047913 - cp's OEIS frontend

A002572 Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.

1, 1, 1, 2, 3, 5, 9, 16, 28, 50, 89, 159, 285, 510, 914, 1639, 2938, 5269, 9451, 16952, 30410, 54555, 97871, 175586, 315016, 565168, 1013976, 1819198, 3263875, 5855833, 10506175, 18849555, 33818794, 60675786, 108861148, 195312750, 350419594, 628704034, 1127987211, 2023774607, 3630948907

Offset: 1

N. J. A. Sloane

This is similar to a question about Egyptian fractions, except that there the denominators of the fractions must be distinct. - N. J. A. Sloane, Jan 13 2021
v(c, d) is the number of partitions of d into positive integers of the form d = c + c_1 + c_2 + ... + c_n, where c_1 <= 2*c, c_{i+1} <= 2*c_i. See Minc link.
Top row of Table 1 of Elsholtz. [Jonathan Vos Post, Aug 30 2011]
a(n+1) is the number of compositions n = p(1) + p(2) + ... + p(m) with p(1)=1 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]
Over an algebraically closed field of characteristic 2, a(n) gives dimensions of the generic cohomology groups H^i_gen(SL_2,L(1)) which are isomorphic to algebraic group cohomology groups H^i(SL_2,L(1)^[i]), where ^[i] denotes i-th Frobenius twist. [David I. Stewart, Oct 22 2013]

			{1}; {1/2 + 1/2}; { 1/2 + 1/4 + 1/4 }; { 1/2 + 1/4 + 1/8 + 1/8, 1/4 + 1/4 + 1/4 + 1/4 }; ...
From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(7+1)=16 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 2*p(k+1):
[ 1]  [ 1 1 1 1 1 1 1 ]
[ 2]  [ 1 1 1 1 1 2 ]
[ 3]  [ 1 1 1 1 2 1 ]
[ 4]  [ 1 1 1 2 1 1 ]
[ 5]  [ 1 1 1 2 2 ]
[ 6]  [ 1 1 2 1 1 1 ]
[ 7]  [ 1 1 2 1 2 ]
[ 8]  [ 1 1 2 2 1 ]
[ 9]  [ 1 1 2 3 ]
[10]  [ 1 2 1 1 1 1 ]
[11]  [ 1 2 1 1 2 ]
[12]  [ 1 2 1 2 1 ]
[13]  [ 1 2 2 1 1 ]
[14]  [ 1 2 2 2 ]
[15]  [ 1 2 3 1 ]
[16]  [ 1 2 4 ]
(End)
From _Joerg Arndt_, Dec 26 2012: (Start)
There are a(8)=16 partitions of 1 into 8 powers of 1/2 (dots denote zeros in the tables of multiplicities in the left column)
[ 1]  [ . 1 1 1 1 1 1 2 ]     + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 2/128
[ 2]  [ . 1 1 1 1 . 4 . ]     + 1/2 + 1/4 + 1/8 + 1/16 + 4/64
[ 3]  [ . 1 1 1 . 3 2 . ]     + 1/2 + 1/4 + 1/8 + 3/32 + 2/64
[ 4]  [ . 1 1 . 3 1 2 . ]     + 1/2 + 1/4 + 3/16 + 1/32 + 2/64
[ 5]  [ . 1 1 . 2 4 . . ]     + 1/2 + 1/4 + 2/16 + 4/32
[ 6]  [ . 1 . 3 1 1 2 . ]     + 1/2 + 3/8 + 1/16 + 1/32 + 2/64
[ 7]  [ . 1 . 3 . 4 . . ]     + 1/2 + 3/8 + 4/32
[ 8]  [ . 1 . 2 3 2 . . ]     + 1/2 + 2/8 + 3/16 + 2/32
[ 9]  [ . 1 . 1 6 . . . ]     + 1/2 + 1/8 + 6/16
[10]  [ . . 3 1 1 1 2 . ]     + 3/4 + 1/8 + 1/16 + 1/32 + 2/64
[11]  [ . . 3 1 . 4 . . ]     + 3/4 + 1/8 + 4/32
[12]  [ . . 3 . 3 2 . . ]     + 3/4 + 3/16 + 2/32
[13]  [ . . 2 3 1 2 . . ]     + 2/4 + 3/8 + 1/16 + 2/32
[14]  [ . . 2 2 4 . . . ]     + 2/4 + 2/8 + 4/16
[15]  [ . . 1 5 2 . . . ]     + 1/4 + 5/8 + 2/16
[16]  [ . . . 8 . . . . ]     + 8/8
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 192-194, 307.
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..2000 (first 201 terms from T. D. Noe)
Christian Elsholtz, Clemens Heuberger, and Daniel Krenn, Algorithmic counting of nonequivalent compact Huffman codes, arXiv:1901.11343 [math.CO], 2019.
Christian Elsholtz, Clemens Heuberger, and Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, arXiv:1108.5964v1 [math.CO], Aug 30, 2011. Also IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075.
Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.
Philippe Flajolet and Helmut Prodinger, Level number sequences for trees, Discrete Math., 65 (1987), 149-156.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 200
Edgar N. Gilbert, Codes based on inaccurate source probabilities, IEEE Trans. Inform. Theory, 17 (1971), 304-315.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
Valerie N. P. Ho, Art B. Owen, and Zexin Pan, Quasi-Monte Carlo with one categorical variable, arXiv:2506.16582 [stat.CO], 2025. See pp. 14, 23.
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.
E. Norwood, The Number of Different Possible Compact Codes, IEEE Transactions on Information Theory, Vol. 13, P. 614, 1967.
J. Paschke et al., Computing and estimating the number of n-ary Huffman sequences of a specified length, Discrete Math., 311 (2011), 1-7. (See p. 3.)
Helmut Prodinger, Philippe Flajolet's early work in combinatorics, arXiv:2103.15791 [math.CO], 2021.
N. J. A. Sloane, Richard Guy and the Encyclopedia of Integer Sequences: A Fifty-Year Friendship, Slides of talk at Conference "Celebrating Richard Guy", University of Calgary, October 2, 2020.
D. I. Stewart, Unbounding Ext, J. Algebra, 365 (2012), 1-11. (See p. 7)
Paul R. Stein, Letter to N. J. A. Sloane, Jul 20 1971
Paul R. Stein, Table of first 127 terms
Index entries for "core" sequences
Index entries for sequences related to trees
Index entries for sequences related to rooted trees
Index entries for sequences related to Egyptian fractions

Cf. A002573, A002574, A007178, A047913, A049284, A049285, A102375, A294775.

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; [ seq(v(1,n), n=1..50) ];

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}]]]; a[n_] := v[1, n-1]; a[1] = 1; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Oct 19 2011, after Maple *)

v(c,d) = if ( d<0 || c<0, 0, if ( d==c, 1, sum(i=1,2*c, v(i,d-c) ) ) )

/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=2;  /* t-ary: t=2 for A002572, t=3 for A176485, t=4 for A176503  */
L=2 + 2*ceil( log(N) / log(t) );
f(k)=(1-t^k)/(1-t);
la(j)=prod(i=1, j, q^f(i) / ( 1 - q^f(i) ) );
nm=sum(j=0, L, (-1)^j * q^f(j) * la(j) );
dn=sum(j=0, L, (-1)^j * la(j) );
gf=nm / dn;
Vec( gf )
/* Joerg Arndt, Dec 27 2012 */

{a(n, k=2) = if( n<2 && k==2, n>=0, nMichael Somos, Dec 20 2016 */

From Jon E. Schoenfield, Dec 18 2016: (Start)
Numerically, it appears that
     lim_{n->infinity} a(n)/c0^n = c1
  and
     lim_{n->infinity} (a(n)/c0^n - c1) / c2^n = c3
where
c0 = 1.79414718754168546349846498809380776370136441826513
       55647141291458811011534167435879115275875728251544
       55034381754309507738861994388752350104180891093803
       27324310643547892073673907996758374498542252887021
       90... = A102375
c1 = 0.14185320208540933707157739062733520381135377764439
       00938624762999524081108574037129602775796177848175
       96757823284956317508884467180902882086032012675483
       68631687927534330190816399081295424373415296405657
       19...
c2 = 0.71317957835995615685267138702642988919007297942329
       35...
c3 = 0.06124104103121269745282188448763211918477582400104
       06... (End)
a(n) = A294775(n-1,1). - Alois P. Heinz, Nov 08 2017

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

A002573 Restricted partitions.

Original entry on oeis.org

0, 1, 1, 2, 4, 7, 12, 22, 39, 70, 126, 225, 404, 725, 1299, 2331, 4182, 7501, 13458, 24145, 43316, 77715, 139430, 250152, 448808, 805222, 1444677, 2591958, 4650342, 8343380, 14969239, 26856992, 48185362, 86451602, 155106844, 278284440, 499283177, 895787396, 1607174300, 2883507098

Offset: 1

N. J. A. Sloane

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=2 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

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

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Shimon Even & 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, A047913, A002574, A049284, A049285.

A column of the triangle in A176431.

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; [ seq(v(2,n), n=1..50) ];

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}]]]; a[n_] := v[2, n]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Jan 30 2012, after Maple *)

A002574 Restricted partitions.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 7, 13, 24, 42, 76, 137, 245, 441, 792, 1420, 2550, 4576, 8209, 14732, 26433, 47424, 85092, 152670, 273914, 491453, 881744, 1581985, 2838333, 5092398, 9136528, 16392311, 29410243, 52766343, 94670652, 169853138, 304741614, 546751437, 980952673, 1759973660

Offset: 1

N. J. A. Sloane

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(8)=13 compositions 8=p(1)+p(2)+...+p(m) with p(1)=3 and p(k) <= 2*p(k+1):
[ 1]  [ 3 1 1 1 1 1 ]
[ 2]  [ 3 1 1 1 2 ]
[ 3]  [ 3 1 1 2 1 ]
[ 4]  [ 3 1 2 1 1 ]
[ 5]  [ 3 1 2 2 ]
[ 6]  [ 3 2 1 1 1 ]
[ 7]  [ 3 2 1 2 ]
[ 8]  [ 3 2 2 1 ]
[ 9]  [ 3 2 3 ]
[10]  [ 3 3 1 1 ]
[11]  [ 3 3 2 ]
[12]  [ 3 4 1 ]
[13]  [ 3 5 ]
(End)

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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, A049284, A049285, A047913.

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; [ seq(v(3,n), n=1..50) ];

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}]]]; a[n_] := v[3, n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 05 2013, after Maple *)

More terms from Michael Somos

A173404 Number of partitions of 1 into up to n powers of 1/2.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 22, 38, 66, 116, 205, 364, 649, 1159, 2073, 3712, 6650, 11919, 21370, 38322, 68732, 123287, 221158, 396744, 711760, 1276928, 2290904, 4110102, 7373977, 13229810, 23735985, 42585540, 76404334, 137080120, 245941268, 441254018, 791673612

Offset: 1

Jonathan Vos Post, Feb 17 2010

Partial sums of number of partitions of 1 into n powers of 1/2. Partial sums of (according to one definition of "binary") the number of binary rooted trees. The subsequence of primes in this partial sum begins: 2, 3, 5, 13, a(43) = 26405436301.

			a(3) = 3: [(1/2)^0], [(1/2)^1,(1/2)^1], [(1/2)^1,(1/2)^2,(1/2)^2].

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Partial sums of A002572.

Cf. A002573, A047913, A002574, A049284, A049285, A007178.

a(n) = Sum_{i=0..n} A002572(i).

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

Original entry on oeis.org

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 *)

cp's OEIS Frontend

A002572 Number of partitions of 1 into n powers of 1/2; or (according to one definition of "binary") the number of binary rooted trees.

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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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A002573 Restricted partitions.

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A002574 Restricted partitions.

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A173404 Number of partitions of 1 into up to n powers of 1/2.

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

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