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 1-6 of 6 results.

A241035 Number of partitions p of n into distinct parts such that max(p) = 2*min(p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 1, 1, 1, 0, 1, 3, 1, 0, 2, 1, 2, 3, 2, 1, 2, 2, 3, 5, 2, 2, 4, 3, 3, 5, 5, 5, 6, 3, 4, 7, 6, 7, 9, 7, 6, 9, 8, 8, 10, 10, 12, 14, 11, 11, 13, 13, 14, 17, 16, 17, 21, 18, 19, 22, 20, 21, 25, 25
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2014

Keywords

Examples

			a(9) counts these 2 partitions:  {6,3}, {4,3,2}.

Crossrefs

Cf. A240874, A241036, A241037.

Programs

  • Mathematica
    z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Min[p]], {n, 0, z}]  (* A240874 *)
Table[Count[f[n], p_ /; Max[p] == 2*Min[p]], {n, 0, z}] (* A241035 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Min[p]], {n, 0, z}] (* A241036 *)
Table[Count[f[n], p_ /; Max[p] > 2*Min[p]], {n, 0, z}]  (* A241037 *)

A240874 Number of partitions p of n into distinct parts such that max(p) < 2*min(p).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 5, 6, 6, 7, 7, 7, 8, 9, 10, 10, 11, 11, 12, 14, 14, 15, 17, 17, 18, 19, 20, 23, 24, 25, 26, 28, 29, 31, 34, 35, 37, 40, 42, 44, 46, 48, 51, 55, 58, 61, 64, 67, 70, 75, 77, 82, 87, 90, 96, 101, 105, 111
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2014

Keywords

Examples

			a(12) counts these 3 partitions:  {12}, {7,5}, {5,4,3}.

Links

Crossrefs

Cf. A241035, A241036, A241037.

Programs

  • Mathematica
    z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Min[p]], {n, 0, z}]  (* this sequence *)
Table[Count[f[n], p_ /; Max[p] == 2*Min[p]], {n, 0, z}] (* A241035 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Min[p]], {n, 0, z}] (* A241036 *)
Table[Count[f[n], p_ /; Max[p] > 2*Min[p]], {n, 0, z}]  (* A241037 *)
  • PARI
    p_q(k) = {prod(j=1,k, 1-q^j);}
GB_q(N,M)= {p_q(N+M)/(p_q(M)*p_q(N));}
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=i, N-(i*(i+1)/2), q^((i/2)*(i+(2*j)-1)) * GB_q(i-1,j-i))));
concat([0],Vec(g))}
A_q(71) \\ John Tyler Rascoe, Mar 16 2024

Formula

G.f.: Sum_{i>0} Sum_{j>=i} q^((i/2)*(i+(2*j)-1)) * q_binomial(i-1,j-i). - John Tyler Rascoe, Mar 16 2024

A241036 Number of partitions p of n into distinct parts such that max(p) >= 2*min(p).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 3, 4, 6, 8, 9, 12, 15, 19, 23, 28, 33, 42, 49, 58, 70, 82, 97, 115, 134, 156, 182, 212, 245, 285, 328, 376, 434, 497, 568, 651, 742, 845, 962, 1090, 1236, 1401, 1584, 1788, 2019, 2273, 2556, 2875, 3227, 3618, 4055, 4538, 5074, 5670, 6327
Offset: 0

Views

Author

Clark Kimberling, Apr 15 2014

Keywords

Examples

			a(11) counts these 9 partitions:  {10,1}, {9,2}, {8,3}, {8,2,1}, {7,3,1}, {6,4,1}, {6,3,2}, {5,4,2}, {5,3,2,1}.

Crossrefs

Cf. A240874, A241035, A241037.

Programs

  • Mathematica
    z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; Max[p] < 2*Min[p]], {n, 0, z}]  (* A240874 *)
Table[Count[f[n], p_ /; Max[p] == 2*Min[p]], {n, 0, z}] (* A241035 *)
Table[Count[f[n], p_ /; Max[p] >= 2*Min[p]], {n, 0, z}] (* A241036 *)
Table[Count[f[n], p_ /; Max[p] > 2*Min[p]], {n, 0, z}]  (* A241037 *)

A241061 Number of partitions p of n into distinct parts such that max(p) < 1 + 2*min(p).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 3, 4, 4, 4, 5, 4, 6, 7, 6, 6, 8, 8, 9, 10, 10, 10, 12, 12, 14, 16, 14, 16, 18, 18, 20, 22, 23, 24, 26, 26, 28, 32, 32, 35, 38, 38, 40, 44, 45, 48, 52, 54, 58, 62, 62, 66, 71, 74, 78, 84, 86, 92, 98, 100, 106, 112, 116, 122
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2014

Keywords

Examples

			a(10) counts these 2 partitions: {10}, {6,4}.

Crossrefs

Cf. A207642, A241062, A241037, A241064, A000009.

Programs

  • Mathematica
    z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
  Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}] (* A241061 *)
  Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}](* A207642 *)
  Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}](* A241062 *)
  Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}](* A241037 *)
  Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}] (* A241064 *)

Formula

a(n) + A241062(n) + A241064(n) = A000009(n) for n >= 1.
a(n) = A207642(n) - A241062(n) for n >= 0.

A241062 Number of partitions p of n into distinct parts such that max(p) = 1 + 2*min(p).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 1, 2, 1, 2, 0, 1, 3, 2, 2, 2, 1, 2, 4, 4, 2, 3, 2, 3, 6, 4, 4, 6, 4, 4, 5, 6, 8, 8, 7, 6, 8, 7, 8, 12, 10, 10, 13, 12, 11, 12, 12, 14, 18, 18, 17, 18, 18, 18, 22, 20, 22, 26, 25, 28, 30, 29, 30, 32
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2014

Keywords

Examples

			a(10) counts these 2 partitions:  73, 532.

Crossrefs

Cf. A207642, A241061, A241037, A241064, A000009.

Programs

  • Mathematica
    z = 70; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
  Table[Count[f[n], p_ /; Max[p] < 1 + 2*Min[p]], {n, 0, z}] (* A241061 *)
  Table[Count[f[n], p_ /; Max[p] <= 1 + 2*Min[p]], {n, 0, z}](* A207642 *)
  Table[Count[f[n], p_ /; Max[p] == 1 + 2*Min[p]], {n, 0, z}](* A241062 *)
  Table[Count[f[n], p_ /; Max[p] >= 1 + 2*Min[p]], {n, 0, z}](* A241037 *)
  Table[Count[f[n], p_ /; Max[p] > 1 + 2*Min[p]], {n, 0, z}] (* A241064 *)

Formula

a(n) + A241061(n) + A241064(n) = A000009(n) for n >= 1.
a(n) = A241037(n) - A241064(n) = A207642(n) - A241061(n) for n >= 0.

A241064 Number of partitions p of n into distinct parts such that max(p) > 1 + 2*min(p).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 4, 6, 8, 11, 13, 16, 21, 26, 32, 38, 45, 56, 66, 79, 94, 110, 128, 151, 178, 207, 240, 277, 320, 370, 426, 488, 561, 642, 732, 834, 948, 1079, 1225, 1388, 1570, 1774, 2002, 2254, 2540, 2856, 3206, 3598, 4034, 4516, 5050, 5642, 6298
Offset: 0

Views

Author

Clark Kimberling, Apr 16 2014

Keywords

Examples

			a(10) counts these 6 partitions:  91, 82, 721, 631, 541, 4321

Crossrefs

Cf. A207642, A241061, A241062, A241064, A000009.

Programs

Formula

a(n) + A241061(n) + A241062(n) = A000009(n) for n >= 1.
a(n) = A241037(n) - A241062(n) for n>= 0.
Showing 1-6 of 6 results.

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