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.

Showing 1-8 of 8 results.

A183568 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n containing a clique of size k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 2, 0, 1, 5, 3, 2, 0, 1, 7, 6, 2, 1, 0, 1, 11, 7, 3, 2, 1, 0, 1, 15, 13, 5, 3, 1, 1, 0, 1, 22, 16, 9, 3, 3, 1, 1, 0, 1, 30, 25, 10, 6, 3, 2, 1, 1, 0, 1, 42, 33, 16, 8, 5, 3, 2, 1, 1, 0, 1, 56, 49, 23, 13, 6, 5, 2, 2, 1, 1, 0, 1, 77, 61, 31, 15, 10, 5, 5, 2, 2, 1, 1, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Jan 05 2011

Keywords

Comments

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique. Each partition has a clique of size 0.

Examples

			T(5,2) = 2, because 2 (of 7) partitions of 5 contain (at least) one clique of size 2: [1,2,2], [1,1,3].
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  1, 1;
   3,  2, 0, 1;
   5,  3, 2, 0, 1;
   7,  6, 2, 1, 0, 1;
  11,  7, 3, 2, 1, 0, 1;
  15, 13, 5, 3, 1, 1, 0, 1;

Links

Crossrefs

Columns k=0-10 give: A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Differences between columns 0 and k (0A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644, A184645.
T(2*k+1,k+1) gives A002865.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=k, [l[1]$2], l))(b(n-i*j, i-1, k)), j=0..n/i)))
    end:
T:= (n, k)-> (l-> l[`if`(k=0, 1, 2)])(b(n, n, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
  • Mathematica
    b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == k, {l[[1]], l[[1]]}, l]][b[n - i*j, i-1, k]], {j, 0, n/i}]] ]; t[n_, k_] := Function[l, l[[If[k == 0, 1, 2]]]][b[n, n, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

Formula

G.f. of column k: (1-Product_{j>0} (1-x^(k*j)+x^((k+1)*j))) / (Product_{j>0} (1-x^j)).

A183567 Number of partitions of n containing a clique of size 10.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 15, 22, 26, 37, 45, 61, 74, 99, 120, 157, 192, 247, 299, 381, 462, 580, 703, 874, 1055, 1303, 1569, 1921, 2309, 2808, 3363, 4070, 4859, 5848, 6964, 8342, 9903, 11817, 13988, 16623, 19626, 23240, 27363, 32297
Offset: 10

Views

Author

Alois P. Heinz, Jan 05 2011

Keywords

Comments

All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.

Examples

			a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].

Links

Crossrefs

10th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=10..60);
  • Mathematica
    max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* Jean-François Alcover, Oct 01 2014 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Length/@Split[#],10]&]],{n,10,60}] (* Harvey P. Dale, Oct 02 2021 *)

Formula

G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).
a(n) = A000041(n) - A184645(n). - Vaclav Kotesovec, Jun 12 2025

A184639 Number of partitions of n having no parts with multiplicity 4.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 14, 19, 27, 37, 50, 67, 88, 115, 153, 196, 253, 324, 412, 524, 661, 828, 1036, 1290, 1603, 1980, 2443, 2997, 3671, 4487, 5460, 6631, 8034, 9703, 11703, 14075, 16890, 20226, 24175, 28838, 34332, 40801, 48394, 57307, 67765, 79974
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Examples

			a(4) = 4, because 4 partitions of 4 have no parts with multiplicity 4: [1,1,2], [2,2], [1,3], [4].

Links

Crossrefs

Cf. A000041, A183561, A183568, A007690, A116645, A118807, A184640, A184641, A184642, A184643, A184644, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[l, If[j == 4, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) - A183561(n).
a(n) = A183568(n,0) - A183568(n,4).
G.f.: Product_{j>0} (1-x^(4*j)+x^(5*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-4*x) + exp(-6*x)) dx = 0.77101366877372033648945034346499691865027592089088481444183... - Vaclav Kotesovec, Jun 12 2025

A184640 Number of partitions of n having no parts with multiplicity 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 14, 21, 28, 39, 51, 72, 92, 124, 160, 210, 266, 349, 438, 562, 704, 892, 1107, 1395, 1720, 2141, 2631, 3249, 3965, 4873, 5916, 7216, 8730, 10585, 12742, 15387, 18443, 22151, 26466, 31646, 37659, 44873, 53212, 63149, 74666, 88295
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Examples

			a(5) = 6, because 6 partitions of 5 have no parts with multiplicity 5: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].

Links

Crossrefs

Cf. A000041, A183562, A183568, A007690, A116645, A118807, A184639, A184641, A184642, A184643, A184644, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 5, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) - A183562(n).
a(n) = A183568(n,0) - A183568(n,5).
G.f.: Product_{j>0} (1-x^(5*j)+x^(6*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-5*x) + exp(-7*x)) dx = 0.78834765570757713777493985857868631321765157344539753651545... - Vaclav Kotesovec, Jun 12 2025

A184641 Number of partitions of n having no parts with multiplicity 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 29, 40, 54, 72, 96, 127, 166, 216, 279, 358, 457, 580, 735, 924, 1159, 1446, 1799, 2228, 2752, 3388, 4158, 5087, 6207, 7551, 9165, 11093, 13401, 16144, 19412, 23286, 27882, 33310, 39727, 47289, 56191, 66647, 78923, 93299
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Examples

			a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].

Links

Crossrefs

Cf. A000041, A183563, A183568, A007690, A116645, A118807, A184639, A184640, A184642, A184643, A184644, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 6, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) - A183563(n).
a(n) = A183568(n,0) - A183568(n,6).
G.f.: Product_{j>0} (1-x^(6*j)+x^(7*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-6*x) + exp(-8*x)) dx = 0.79818518024793359047735154473665146019665210453617381247423... - Vaclav Kotesovec, Jun 12 2025

A184642 Number of partitions of n having no parts with multiplicity 7.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 22, 29, 41, 54, 75, 97, 130, 168, 222, 283, 368, 465, 597, 750, 949, 1183, 1488, 1841, 2292, 2822, 3487, 4267, 5239, 6376, 7782, 9429, 11439, 13798, 16661, 20007, 24043, 28763, 34420, 41021, 48894, 58066, 68956, 81627, 96592
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Examples

			a(7) = 14, because 14 partitions of 7 have no parts with multiplicity 7: [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,1,3], [1,1,2,3], [2,2,3], [1,3,3], [1,1,1,4], [1,2,4], [3,4], [1,1,5], [2,5], [1,6], [7].

Links

Crossrefs

Cf. A000041, A183564, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184643, A184644, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=7, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 7, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(FreeQ[Length/@Split[#],7]&)],{n,0,50}] (* _Harvey P. Dale, Sep 21 2024 *)

Formula

a(n) = A000041(n) - A183564(n).
a(n) = A183568(n,0) - A183568(n,7).
G.f.: Product_{j>0} (1-x^(7*j)+x^(8*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-7*x) + exp(-9*x)) dx = 0.80430417180776436899064351977235191494130305607975798117531... - Vaclav Kotesovec, Jun 12 2025

A184643 Number of partitions of n having no parts with multiplicity 8.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 30, 41, 55, 75, 99, 131, 172, 223, 288, 372, 474, 603, 764, 962, 1206, 1509, 1876, 2326, 2878, 3543, 4351, 5330, 6506, 7921, 9623, 11655, 14085, 16987, 20434, 24529, 29392, 35138, 41930, 49947, 59381, 70474, 83512, 98779
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Links

Crossrefs

Cf. A000041, A183565, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184644, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=8, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 8, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) - A183565(n).
a(n) = A183568(n,0) - A183568(n,8).
G.f.: Product_{j>0} (1-x^(8*j)+x^(9*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-8*x) + exp(-10*x)) dx = 0.80836901097063952622501649557292291036896118821761722817375... - Vaclav Kotesovec, Jun 12 2025

A184644 Number of partitions of n having no parts with multiplicity 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 42, 55, 76, 99, 133, 172, 227, 290, 376, 477, 612, 769, 975, 1217, 1528, 1895, 2359, 2907, 3592, 4400, 5403, 6584, 8034, 9742, 11823, 14272, 17234, 20713, 24897, 29803, 35674, 42542, 50719, 60272, 71592, 84794
Offset: 0

Views

Author

Alois P. Heinz, Jan 18 2011

Keywords

Links

Crossrefs

Cf. A000041, A183566, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184645.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=9, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 9, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

Formula

a(n) = A000041(n) - A183566(n).
a(n) = A183568(n,0) - A183568(n,9).
G.f.: Product_{j>0} (1-x^(9*j)+x^(10*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-9*x) + exp(-11*x)) dx = 0.81120660452677002313966407107688916817839171627473737415672... - Vaclav Kotesovec, Jun 12 2025
Showing 1-8 of 8 results.

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