cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A356846 Number of integer compositions of n into parts not covering an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 2, 5, 11, 25, 57, 115, 236, 482, 978, 1986, 4003, 8033, 16150, 32402, 64943, 130207, 260805, 522123, 1045168, 2091722, 4185431, 8374100, 16753538, 33515122, 67042865, 134106640, 268246886, 536549760, 1073194999, 2146553011, 4293391411, 8587283895
Offset: 0

Views

Author

Gus Wiseman, Sep 03 2022

Keywords

Examples

			The a(0) = 0 through a(6) = 8 compositions:
  .  .  .  .  (13)  (14)   (15)
              (31)  (41)   (24)
                    (113)  (42)
                    (131)  (51)
                    (311)  (114)
                           (141)
                           (411)
                           (1113)
                           (1131)
                           (1311)
                           (3111)

Crossrefs

The complement is counted by A107428, initial A107429.
The case of partitions is A239955, ranked by A073492, initial A053251, complement A034296.
These compositions are ranked by A356842, complement A356841.
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A060142, A080259, A188575, A239327, A356604, A356605.

Programs

  • Mathematica
    gappyQ[m_]:=And[m!={},Union[m]!=Range[Min[m],Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],gappyQ]],{n,0,15}]

Formula

a(n) = A011782(n) - A107428(n).

A304706 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 6, 5, 6, 6, 8, 7, 11, 10, 11, 12, 15, 14, 18, 17, 20, 23, 27, 25, 31, 32, 35, 38, 43, 43, 51, 54, 59, 63, 71, 73, 85, 89, 96, 102, 113, 120, 134, 141, 149, 161, 175, 183, 203, 213, 233, 252, 280, 293, 319, 338, 360, 383, 409, 430, 468, 493, 531, 565
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (3, 3)                      | (3, 3/2)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Crossrefs

Cf. A053251, A053282, A304705, A304707, A304708.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

a(n) <= A304705(n).

A304707 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and d1 < d2 < ... < dm.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 4, 5, 7, 5, 7, 8, 8, 10, 12, 10, 11, 14, 14, 14, 18, 17, 20, 23, 22, 26, 30, 29, 32, 35, 34, 37, 43, 44, 48, 54, 54, 59, 67, 70, 76, 81, 84, 89, 97, 101, 110, 119, 123, 129, 139, 145, 155, 171, 176, 189, 201, 211, 228, 245, 257, 274, 295
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (1, 2, 3)                   | (1, 1, 1)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)

Crossrefs

Cf. A053251, A053282, A304705, A304706, A304708.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t) +`if`(i/t>r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

A304708 Number of partitions (d1,d2,...,dm) of n such that d1/1 > d2/2 > ... > dm/m and d1 < d2 < ... < dm.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 3, 5, 5, 4, 5, 6, 6, 7, 8, 8, 9, 10, 12, 11, 13, 13, 16, 16, 15, 18, 21, 22, 26, 25, 28, 31, 33, 33, 35, 39, 41, 46, 47, 50, 53, 59, 63, 68, 74, 77, 84, 90, 93, 98, 105, 111, 119, 129, 132, 138, 149, 157, 169, 178, 189, 201, 211, 227
Offset: 0

Views

Author

Seiichi Manyama, May 17 2018

Keywords

Examples

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+-------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
3 | (3)                         | (3)
4 | (4)                         | (4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
6 | (6)                         | (6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
9 | (9)                         | (9)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)

Crossrefs

Cf. A053251, A053282, A304705, A304706, A304707.

Programs

  • Maple
    b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>=r, 0, b(n-i, i/t, i+1, t+1))))
    end:
a:= n-> b(n, n+1, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018
  • Mathematica
    b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t >= r, 0, b[n - i, i/t, i + 1, t + 1]]]];
a[n_] := b[n, n + 1, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

a(n) <= A304707(n).

A351594 Number of odd-length integer partitions y of n that are alternately constant, meaning y_i = y_{i+1} for all odd i.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 2, 7, 3, 9, 4, 13, 6, 19, 6, 26, 10, 35, 12, 49, 16, 64, 20, 87, 27, 115, 32, 151, 44, 195, 53, 256, 69, 328, 84, 421, 108, 537, 130, 682, 167, 859, 202, 1085, 252, 1354, 305, 1694, 380, 2104, 456, 2609, 564, 3218, 676, 3968, 826, 4863
Offset: 0

Views

Author

Gus Wiseman, Feb 24 2022

Keywords

Comments

These are partitions with all even run-lengths except for the last, which is odd.

Examples

			The a(1) = 1 through a(9) = 7 partitions:
  (1)  (2)  (3)    (4)  (5)      (6)    (7)        (8)    (9)
            (111)       (221)    (222)  (331)      (332)  (333)
                        (11111)         (22111)           (441)
                                        (1111111)         (22221)
                                                          (33111)
                                                          (2211111)
                                                          (111111111)

Crossrefs

The ordered version (compositions) is A016116 shifted right once.
All odd-length partitions are counted by A027193.
The opposite version is A117409, even-length A351012, any length A351003.
Replacing equal with unequal relations appears to give:
- any length: A122129
- odd length: A122130
- even length: A351008
- opposite any length: A122135
- opposite odd length: A351595
- opposite even length: A122134
This is the odd-length case of A351004, even-length A035363.
The case that is also strict at even indices is:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
A reverse version is A096441; see also A349060.
Cf. A000009, A000041, A000070, A000984, A003242, A027383, A053738, A236559, A236914, A350842.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]==#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,30}]

A356605 Number of integer compositions of n into odd parts covering an interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 6, 10, 15, 26, 41, 65, 104, 164, 262, 424, 687, 1112, 1792, 2898, 4677, 7556, 12197, 19699, 31836, 51466, 83234, 134593, 217674, 352057, 569452, 921165, 1490173, 2410784, 3900288, 6310436, 10210358, 16521108, 26733020, 43258086, 69999295
Offset: 0

Views

Author

Gus Wiseman, Aug 31 2022

Keywords

Examples

			The a(1) = 1 through a(8) = 15 compositions:
  (1)  (11)  (3)    (13)    (5)      (33)      (7)        (35)
             (111)  (31)    (113)    (1113)    (133)      (53)
                    (1111)  (131)    (1131)    (313)      (1133)
                            (311)    (1311)    (331)      (1313)
                            (11111)  (3111)    (11113)    (1331)
                                     (111111)  (11131)    (3113)
                                               (11311)    (3131)
                                               (13111)    (3311)
                                               (31111)    (111113)
                                               (1111111)  (111131)
                                                          (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)

Crossrefs

These compositions are ranked by the intersection of A060142 and A356841.
Before restricting to odds we have A107428, initial A107429.
The not necessarily gapless version is A324969 (essentially A000045).
The strict case is A332032.
The initial case is A356604.
The case of partitions is A356737, initial A053251 (ranked by A356232).
A000041 counts partitions, compositions A011782.
A066208 lists numbers with all odd prime indices, counted by A000009.
A073491 lists numbers with gapless prime indices, initial A055932.
Cf. A001227, A066205, A137921, A333217, A356224, A356846.

Programs

  • Mathematica
    nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], And@@OddQ/@#&&nogapQ[(#+1)/2]&]],{n,0,15}]

Extensions

More terms from Alois P. Heinz, Sep 01 2022

A376628 G.f.: Sum_{k>=0} x^(k*(k+1)) / Product_{j=1..k} (1 - x^(2*j-1)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 8, 10, 10, 12, 14, 15, 17, 19, 21, 23, 27, 29, 31, 37, 39, 43, 49, 52, 58, 64, 70, 76, 84, 92, 99, 111, 119, 129, 143, 153, 167, 183, 197, 213, 233, 251, 271, 295, 317, 343, 372, 400, 430, 466, 500, 538, 582, 622, 670
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A003106, A053251, A053254, A053258, A053282, A333179.

Programs

  • Mathematica
    nmax=100; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} x^(2*j)/(1 - x^(2*j-1)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(5/2) * sqrt(n)).
Conjectural g.f.: (1 + q * nu(-q))/(1 + q) = 1 + Sum_{k >= 0} q^(k+2)*Product_{j = 1..k} 1 + q^(2*j+1), where nu(q) is the g.f. of A053254. - Peter Bala, Jan 03 2025

A376629 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} (1 + x^(2*j-1)).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 2, 2, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 3, 3, 1, 3, 3, 3, 2, 2, 3, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 4, 6, 6, 4, 6, 5, 7, 6, 5, 7, 7, 6, 6, 8, 7, 7, 7, 9, 8
Offset: 0

Views

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

Links

Crossrefs

Cf. A003106, A053251, A053258, A333179, A376630.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Sum[x^(k*(k+1))*Product[1+x^(2*j-1), {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 + x^(2*k - 1))*x^(2*k)]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += p;, {k, 1, Sqrt[nmax]}]; Take[CoefficientList[s, x], nmax + 1]

Formula

G.f.: Sum_{k>=0} Product_{j=1..k} (x^(2*j) + x^(4*j-1)).
a(n) ~ exp(Pi*sqrt(n/30)) / (2*5^(1/4)*sqrt(n)).

A109471 Cumulative sum of absolute values of coefficients of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

Original entry on oeis.org

1, 3, 6, 11, 17, 27, 38, 55, 76, 103, 136, 182, 235, 303, 385, 489, 612, 766, 945, 1166, 1428, 1742, 2111, 2557, 3072, 3686, 4401, 5246, 6223, 7371, 8692, 10236, 12014, 14074, 16435, 19171, 22292, 25884, 29981, 34677, 40017, 46122, 53038, 60920
Offset: 0

Views

Author

Jonathan Vos Post, Aug 28 2005

Keywords

Links

Crossrefs

Cf. A000039, A000019, A053250, A053251, A053252, A053253, A053254, A053255, A064053.

Programs

  • Mathematica
    nmax = 200; f[q_, s_] := Sum[q^(n^2)/Product[1 + q^k, {k, n}]^2, {n, 0, s}]; A000039:= CoefficientList[Series[f[q, nmax], {q, 0, nmax}], q][[1 ;; -1 ;; 2]]; Table[Sum[Abs[A000039[[k]]], {k,1,n}], {n,1,51}] (* G. C. Greubel, Feb 18 2018 *)

Formula

a(n) = Sum_{k=0..n} abs(A000039(k)). [corrected by Joerg Arndt, Feb 25 2018]
a(n) ~ sqrt(3/2) * exp(sqrt(n/3)*Pi) / Pi. - Vaclav Kotesovec, Jun 12 2019

A192432 Coefficients of a mock theta function.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 8, 11, 13, 17, 22, 27, 34, 42, 51, 62, 76, 91, 109, 132, 156, 186, 221, 259, 306, 360, 420, 490, 572, 663, 769, 892, 1027, 1184, 1364, 1564, 1793, 2053, 2343, 2674, 3048, 3464, 3935, 4465, 5056, 5721, 6468, 7297, 8227, 9269, 10423
Offset: 0

Views

Author

Jeremy Lovejoy, Jun 30 2011

Keywords

Comments

Essentially the unified WRT invariant of the Seifert manifold M(2,3,8)

Links

Crossrefs

a(n) equals A053251(2n+2).

Programs

  • PARI
    N=66; q='q+O('q^N); gf=sum(n=0,N,q^n*prod(k=1,2*n+1,1+q^k)); Vec(gf) \\ Joerg Arndt, Jul 01 2011

Formula

a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(2*n)). - Vaclav Kotesovec, Jun 12 2019
Previous Showing 51-60 of 66 results. Next

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