A176503 -id:A176503 - 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

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 *)

A176485 First column of triangle in A176452.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 25, 48, 92, 176, 338, 649, 1246, 2392, 4594, 8823, 16945, 32545, 62509, 120060, 230598, 442910, 850701, 1633948, 3138339, 6027842, 11577747, 22237515, 42711863, 82037200, 157569867, 302646401, 581296715, 1116503866, 2144482948, 4118935248, 7911290530

Offset: 1

N. J. A. Sloane, Dec 07 2010

nonn

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

Row 2 of Table 1 of Elsholtz, row 1 being A002572. - Jonathan Vos Post, Aug 30 2011

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(7+1)=25 compositions 7=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 3*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 1 3 ]
[ 5]  [ 1 1 1 2 1 1 ]
[ 6]  [ 1 1 1 2 2 ]
[ 7]  [ 1 1 1 3 1 ]
[ 8]  [ 1 1 2 1 1 1 ]
[ 9]  [ 1 1 2 1 2 ]
[10]  [ 1 1 2 2 1 ]
[11]  [ 1 1 2 3 ]
[12]  [ 1 1 3 1 1 ]
[13]  [ 1 1 3 2 ]
[14]  [ 1 2 1 1 1 1 ]
[15]  [ 1 2 1 1 2 ]
[16]  [ 1 2 1 2 1 ]
[17]  [ 1 2 1 3 ]
[18]  [ 1 2 2 1 1 ]
[19]  [ 1 2 2 2 ]
[20]  [ 1 2 3 1 ]
[21]  [ 1 2 4 ]
[22]  [ 1 3 1 1 1 ]
[23]  [ 1 3 1 2 ]
[24]  [ 1 3 2 1 ]
[25]  [ 1 3 3 ]
(End)

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

Cf. A176452, A002572, A176503, A294775.

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_] := b[2n-1, 1, 3];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

/* g.f. as given in the Elsholtz/Heuberger/Prodinger reference */
N=66;  q='q+O('q^N);
t=3;  /* 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) = A294775(n-1,2). - Alois P. Heinz, Nov 08 2017

Extended by Jonathan Vos Post, Aug 30 2011

Added terms beyond a(20)=62509, Joerg Arndt, Dec 18 2012.

A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 31, 61, 121, 240, 476, 944, 1872, 3712, 7362, 14601, 28958, 57432, 113904, 225904, 448034, 888583, 1762321, 3495200, 6932008, 13748208, 27266738, 54077957, 107252486, 212713209, 421872826, 836697836, 1659417786, 3291113315, 6527245245, 12945446241, 25674625681

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1), see example. - Joerg Arndt, Dec 18 2012

Row 4 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, and row 3 being A176503.

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(6+1)=16 compositions 6=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 5*p(k+1):
[ 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 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, 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.

Cf. A002572, A176485, A176503, A294775.

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_] := b[4n - 3, 1, 5];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=5 */
```

a(n) = A294775(n-1,4). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=113904 added by Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017

A194629 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 63, 125, 249, 496, 988, 1968, 3920, 7808, 15552, 30978, 61705, 122910, 244824, 487664, 971376, 1934880, 3854082, 7676935, 15291665, 30459424, 60672040, 120852464, 240725680, 479500802, 955116293, 1902493446, 3789571321, 7548436410

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 6*p(k+1). - Joerg Arndt, Dec 18 2012

Row 5 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, and row 4 being A194628.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, 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.

Cf. A002572, A176485, A176503, A194628, A294775.

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_] := b[5n-4, 1, 6];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=6 */
```

a(n) = A294775(n-1,5). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=122910 added by Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017

A194630 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, 505, 1008, 2012, 4016, 8016, 16000, 31936, 63744, 127234, 253961, 506910, 1011800, 2019568, 4031088, 8046112, 16060160, 32056322, 63984903, 127714833, 254920736, 508825640, 1015623664, 2027200176, 4046322176, 8076520194

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 7*p(k+1). - Joerg Arndt, Dec 18 2012

Row 6 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, and row 5 being A194629.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, 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.

Cf. A002572, A176485, A176503, A194628, A194629, A294775.

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_] := b[6n-5, 1, 7];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=7 */
```

a(n) = A294775(n-1,6). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=127234 added by Joerg Arndt, Dec 18 2012

A194631 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 255, 509, 1017, 2032, 4060, 8112, 16208, 32384, 64704, 129280, 258304, 516098, 1031177, 2060318, 4116568, 8225008, 16433776, 32835104, 65605376, 131081216, 261903618, 523290119, 1045547025, 2089029664, 4173934632, 8339628016

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 8*p(k+1). - Joerg Arndt, Dec 18 2012

Row 7 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, row 5 being A194629, and row 6 being A194630.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Christian Elsholtz, Clemens Heuberger, 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.

Cf. A002572, A176485, A176503, A194628, A194629, A194630, A294775.

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_] := b[7n-6, 1, 8];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=8 */
```

a(n) = A294775(n-1,7). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=129280 added by Joerg Arndt, Dec 18 2012

A194633 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1023, 2045, 4089, 8176, 16348, 32688, 65360, 130688, 261312, 522496, 1044736, 2088960, 4176896, 8351746, 16699401, 33390622, 66764888, 133497072, 266928752, 533726752, 1067192064, 2133861376, 4266677504, 8531265024

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 10*p(k+1). [Joerg Arndt, Dec 18 2012]

Row 9 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, row 5 being A194629, row 6 being A194630, row 7 being A194631, and row 8 being A194632.

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

Cf. A002572, A176485, A176503, A194628 - A194632, A294775.

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_] := b[9n-8, 1, 10];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=10 */
```

a(n) = A294775(n-1,9).

Added terms beyond a(20)=130688, Joerg Arndt, Dec 18 2012

Invalid empirical g.f. removed by Alois P. Heinz, Nov 08 2017

A176463 Irregular triangle read by rows: T(n,k) = number of Huffman-equivalence classes of ternary trees with 3n+1 leaves and 4k leaves on the bottom level (n>=1, k>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 8, 4, 2, 1, 15, 8, 4, 2, 29, 15, 8, 4, 1, 57, 29, 15, 8, 2, 1, 112, 57, 29, 15, 4, 2, 1, 220, 112, 57, 29, 7, 4, 2

Offset: 1

N. J. A. Sloane, Dec 07 2010

			Triangle begins:
1
1
1 1
2 1 1
4 2 1 1
8 4 2 1
15 8 4 2
29 15 8 4 1
57 29 15 8 2 1
112 57 29 15 4 2 1
220 112 57 29 7 4 2

Christian Elsholtz, Clemens Heuberger and Helmut Prodinger, The number of Huffman codes, compact trees, and sums of unit fractions, IEEE Trans. Information Theory, Vol. 59, No. 2, 2013 pp. 1065-1075; also arXiv:1108.5964 [math.CO], 2011.
Jordan Paschke, Jeffrey Burkert and Rebecca Fehribach, Computing and estimating the number of n-ary Huffman sequences of a specified length, Discrete Math., 311 (2011), 1-7.

Cf. A176431, A176452, A194628 - A194633. Leading column gives A176503.

A194632 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2041, 4080, 8156, 16304, 32592, 65152, 130240, 260352, 520448, 1040384, 2079746, 4157449, 8310814, 16613464, 33210608, 66388592, 132711968, 265293568, 530326528, 1060132096, 2119222786, 4236363783, 8468566033

Offset: 1

Jonathan Vos Post, Aug 30 2011

nonn

a(n+1) is the number of compositions n=p(1)+p(2)+...+p(m) with p(1)=1 and p(k) <= 9*p(k+1). - Joerg Arndt, Dec 18 2012

Row 8 of Table 1 of Elsholtz, row 1 being A002572, row 2 being A176485, row 3 being A176503, row 4 being A194628, row 5 being A194629, row 6 being A194630, and row 7 being A194631.

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

Cf. A002572, A176485, A176503, A194628, A194629, A194630, A194631, A294775.

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_] := b[8n-7, 1, 9];
Array[a, 40] (* Jean-François Alcover, Jul 21 2018, after Alois P. Heinz *)

PARI
```
/* see A002572, set t=9 */
```

a(n) = A294775(n-1,8). - Alois P. Heinz, Nov 08 2017

Terms beyond a(20)=130240 added by Joerg Arndt, Dec 18 2012

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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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.

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A176485 First column of triangle in A176452.

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A194628 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

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A194629 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

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A194630 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

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A194631 Arises in enumerating Huffman codes, compact trees, and sums of unit fractions.

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