A002572 -id:A002572 - cp's OEIS frontend

A049284 Restricted partitions.

0, 0, 0, 1, 1, 2, 4, 7, 13, 24, 43, 78, 140, 251, 452, 812, 1457, 2617, 4697, 8428, 15126, 27142, 48700, 87384, 156787, 281307, 504723, 905562, 1624731, 2915039, 5230040, 9383505, 16835453, 30205347, 54192931, 97230224, 174445475, 312981054, 561534340, 1007475560

Offset: 1

N. J. A. Sloane, Michael Somos

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=4 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

			From _Joerg Arndt_, Dec 18 2012: (Start)
There are a(9)=13 compositions 9=p(1)+p(2)+...+p(m) with p(1)=4 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)

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.

Shimon Even and Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.

Cf. A002572, A002573, A002574, A049285.

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(4,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}]]]; Table[v[4, n], {n, 1, 40}] (* Jean-François Alcover, Jan 10 2014, translated from Maple *)

A049285 Restricted partitions.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 7, 13, 24, 43, 78, 141, 253, 455, 818, 1468, 2637, 4734, 8495, 15247, 27361, 49094, 88093, 158063, 283599, 508840, 912956, 1638003, 2938861, 5272795, 9460227, 16973125, 30452380, 54636174, 98025512, 175872397, 315541228, 566127763

Offset: 1

N. J. A. Sloane, Michael Somos

Number of compositions n=p(1)+p(2)+...+p(m) with p(1)=5 and p(k) <= 2*p(k+1), see example. [Joerg Arndt, Dec 18 2012]

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

Shimon Even & Abraham Lempel, Generation and enumeration of all solutions of the characteristic sum condition, Information and Control 21 (1972), 476-482.

Cf. A002572, A002573, A049284, A002574.

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(5,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}]]]; Table[v[5, n], {n, 1, 40}] (* Jean-François Alcover, Jan 10 2014, translated from Maple *)

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

A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

Original entry on oeis.org

0, 0, 0, 2, 7, 52, 297, 1994, 14594, 113794, 991741, 9199390, 94105010, 1015012796, 11914379971, 146974330141, 1954701366709

Offset: 1

N. J. A. Sloane, Dec 15 2016

			Richard Guy gives examples in his letter.

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.
Nowakowski, Richard Joseph, Generalization of the Langford-Skolem problem, MS Thesis, University of Calgary, 1975.

R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence "J".
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

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

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).

A176431 Irregular triangle read by rows: T(n,k) = number of Huffman-equivalence classes of binary trees with n leaves and 2k leaves on the bottom level (n>=2, k>=1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 5, 3, 1, 9, 5, 1, 1, 16, 9, 2, 1, 28, 16, 4, 2, 50, 28, 7, 4, 89, 50, 12, 7, 1, 159, 89, 22, 12, 2, 1, 285, 159, 39, 22, 3, 2, 510, 285, 70, 39, 22, 3, 1

Offset: 2

N. J. A. Sloane, Dec 07 2010

			Triangle begins:
1
1
1 1
2 1
3 2
5 3 1
9 5 1 1
16 9 2 1
28 16 4 2
50 28 7 4
89 50 12 7 1
159 89 22 12 2 1
285 159 39 22 3 2
510 285 70 39 22 3 1

J. Paschke et al., Computing and estimating the number of n-ary Huffman sequences of a specified length, Discrete Math., 311 (2011), 1-7.

Cf. A176452, A176463. First three columns are A002572 (twice), A002573.

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

A049284 Restricted partitions.

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A049285 Restricted partitions.

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Maple

Mathematica

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

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Extensions

A279198 Number of pairs of conjugate inseparable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

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Formula

Extensions

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

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Author

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Comments

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Programs

Mathematica

PARI

Formula

Extensions

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

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

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Author

Keywords

Comments

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Programs

Mathematica