A207642 -id:A207642 - cp's OEIS frontend

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

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

Clark Kimberling, Apr 16 2014

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

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

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 *)

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

Clark Kimberling, Apr 16 2014

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

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

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 *)

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

Clark Kimberling, Apr 16 2014

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

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

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 *)
Table[Count[IntegerPartitions[n],?(Length[#]==Length[Union[#]]&&#[[1]]>2#[[-1]]+1&)],{n,0,60}]//Quiet (* _Harvey P. Dale, Sep 25 2024 *)

a(n) + A241061(n) + A241062(n) = A000009(n) for n >= 1.

a(n) = A241037(n) - A241062(n) for n>= 0.

A384426 G.f.: Sum_{k>=1} x^k * Product_{j=k..2*k} (1 + x^j).

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 4, 4, 4, 5, 6, 5, 6, 7, 8, 8, 9, 9, 10, 12, 12, 13, 14, 14, 16, 18, 19, 20, 21, 22, 24, 26, 28, 30, 32, 33, 34, 38, 40, 43, 46, 48, 51, 54, 56, 60, 64, 67, 72, 77, 80, 84, 88, 92, 98, 105, 110, 116, 122, 128, 134, 142, 148, 155, 164, 172

Offset: 0

Vaclav Kotesovec, Jun 14 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A237824, A207642.

nmax = 100; CoefficientList[Series[Sum[x^k * Product[1 + x^j, {j,k,2*k}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 + x^(2*k - 1))*(1 + x^(2*k))/(1 + x^k)]; p = Normal[p + O[x]^nmax]; s += p*(1 + x^k)*x^k;, {k, 1, nmax}]; Take[CoefficientList[s, x], nmax + 1]

a(n) ~ c * exp(r*sqrt(n)) / sqrt(n), where r = 0.926140105877... = 2*sqrt((3/2)*log(z)^2 - polylog(2, 1-z) + polylog(2, 1-z^2)), where z = (-1 + (44 - 3*sqrt(177))^(1/3) + (44 + 3*sqrt(177))^(1/3))/6 = 0.82948354095849703967... is the real root of the equation z^3*(1 - z)/(1 - z^2)^2 = 1 and c = 0.6975701...

cp's OEIS Frontend

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

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

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

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A384426 G.f.: Sum_{k>=1} x^k * Product_{j=k..2*k} (1 + x^j).

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