A237827 -id:A237827 - cp's OEIS frontend

A118096 Number of partitions of n such that the largest part is twice the smallest part.

0, 0, 1, 1, 2, 3, 3, 4, 6, 6, 6, 10, 9, 11, 13, 14, 15, 20, 18, 23, 25, 27, 27, 37, 35, 39, 43, 48, 49, 61, 57, 68, 72, 78, 81, 97, 95, 107, 114, 127, 128, 150, 148, 168, 179, 191, 198, 229, 230, 254, 266, 291, 300, 338, 344, 379, 398, 427, 444, 498, 505, 550, 580, 625

Offset: 1

Emeric Deutsch, Apr 12 2006

nonn

Also number of partitions of n such that if the largest part occurs k times, then the number of parts is 2k. Example: a(8)=4 because we have [7,1], [6,2], [5,3] and [3,3,1,1].

			a(8)=4 because we have [4,2,2], [2,2,2,1,1], [2,2,1,1,1,1] and [2,1,1,1,1,1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A237825, A237826, A237827.

Cf. A008483, A117086, A238479, A350893.

g:=sum(x^(3*k)/product(1-x^j,j=k..2*k),k=1..30): gser:=series(g,x=0,75): seq(coeff(gser,x,n),n=1..70);
# second Maple program:
b:= proc(n, i, t) option remember: `if`(n=0, 1, `if`(in, 0, b(n-i, i, t))))
    end:
a:= n-> add(b(n-3*j, 2*j, j), j=1..n/3):
seq(a(n), n=1..64);  # Alois P. Heinz, Sep 04 2017

Table[Count[IntegerPartitions[n], p_ /; 2 Min[p] = = Max[p]], {n, 40}] (* Clark Kimberling, Feb 16 2014 *)
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < t, 0,
     b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t]]]];
a[n_] := Sum[b[n - 3j, 2j, j], {j, 1, n/3}];
Array[a, 64] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
(* Third program: *)
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+1)]; s += x^(3*k)/(1 - x^k)/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 16 2025 *)

my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=k, 2*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

G.f.: Sum_{k>=1} x^(3*k)/Product_{j=k..2*k} (1-x^j).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(1/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 13 2025

A237828 Number of partitions of n such that 2*(least part) + 1 = greatest part.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 4, 6, 9, 10, 12, 17, 18, 22, 27, 31, 34, 42, 45, 53, 61, 66, 72, 86, 92, 103, 113, 125, 135, 154, 163, 180, 197, 213, 229, 257, 271, 294, 318, 346, 368, 404, 426, 463, 497, 532, 564, 616, 651, 700, 747, 798, 844, 912, 962, 1033, 1097, 1167, 1231, 1327, 1397, 1486, 1576, 1677

Offset: 1

Clark Kimberling, Feb 16 2014

Also, the number of partitions p of n such that if h = max(p), then h is an (h,0)-separator of p; for example, a(10) counts these 9 partitions: 181, 271, 361, 262, 451, 352, 343, 23131, 1212121. - Clark Kimberling, Mar 24 2014

			a(8) = 4 counts these partitions:  3311, 3221, 32111, 311111.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A118096, A237829.

Cf. A049820, A237757, A237825, A237826, A237827, A000041.

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] = = Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] = = Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] = = Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 = = Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 = = Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n],?(2*Min[#]+1==Max[#]&)],{n,60}] (* _Harvey P. Dale, Jun 25 2017 *)
kmax = 65;
Sum[x^(3k+1)/Product[1-x^j, {j, k, 2k+1}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)
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 + 1)]; s += x^(3*k + 1)/(1 - x^k)/(1 - x^(2*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 18 2025 *)

my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(3*k+1)/prod(j=k, 2*k+1, 1-x^j)))) \\ Seiichi Manyama, May 17 2023

G.f.: Sum_{k>=1} x^(3*k+1)/Product_{j=k..2*k+1} (1-x^j). - Seiichi Manyama, May 17 2023

a(n) ~ sqrt(phi) * exp(Pi*sqrt(2*n/15)) / (sqrt(2)* 5^(1/4) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 20 2025

A237825 Number of partitions of n such that 3*(least part) = greatest part.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 5, 8, 9, 13, 14, 18, 20, 27, 28, 35, 38, 49, 51, 61, 66, 81, 86, 102, 109, 130, 136, 161, 172, 202, 214, 245, 264, 305, 323, 369, 395, 452, 480, 544, 580, 657, 703, 786, 842, 947, 1008, 1124, 1205, 1340, 1432, 1589, 1702, 1886, 2014

Offset: 1

Clark Kimberling, Feb 16 2014

			a(7) = 3 counts these partitions:  331, 3211, 31111.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000041, A117086, A237757, A237828, A237829.

Cf. A118096, A237826, A237827.

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n],?(3#[[-1]]==#[[1]]&)],{n,60}] (* _Harvey P. Dale, May 14 2023 *)
kmax = 57;
Sum[x^(4 k)/Product[1 - x^j, {j, k, 3 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)

my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=k, 3*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

G.f.: Sum_{k>=1} x^(4*k)/Product_{j=k..3*k} (1-x^j). - Seiichi Manyama, May 14 2023

a(n) ~ c * A376815^sqrt(n) / sqrt(n), where c = 0.23036554... - Vaclav Kotesovec, Jun 14 2025

A237826 Number of partitions of n such that 4*(least part) = greatest part.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 26, 31, 38, 47, 55, 67, 78, 92, 106, 126, 145, 167, 190, 219, 247, 288, 320, 366, 410, 466, 520, 591, 654, 739, 820, 924, 1018, 1148, 1263, 1415, 1562, 1740, 1911, 2136, 2342, 2607, 2859, 3169, 3469, 3849, 4208

Offset: 1

Clark Kimberling, Feb 16 2014

			a(8) = 3 counts these partitions:  431, 4211, 41111.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000041, A117086, A237757, A237828, A237829.

Cf. A118096, A237825, A237827.

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
Table[Count[IntegerPartitions[n],?(#[[1]]==4#[[-1]]&)],{n,60}] (* _Harvey P. Dale, Jun 15 2023 *)
kmax = 55;
Sum[x^(5k)/Product[1 - x^j, {j, k, 4 k}], {k, 1, kmax}]/x + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)

my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(5*k)/prod(j=k, 4*k, 1-x^j)))) \\ Seiichi Manyama, May 14 2023

G.f.: Sum_{k>=1} x^(5*k)/Product_{j=k..4*k} (1-x^j). - Seiichi Manyama, May 14 2023

a(n) ~ c * d^sqrt(n) / sqrt(n), where d = 4.9219345... and c = 0.1699648... - Vaclav Kotesovec, Jun 19 2025

A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 59, 73, 94, 117, 148, 181, 228, 277, 344, 418, 514, 621, 762, 917, 1116, 1342, 1624, 1945, 2348, 2803, 3366, 4012, 4798, 5700, 6798, 8052, 9565, 11305, 13383, 15771, 18618, 21880, 25745, 30187, 35414, 41414, 48461, 56531, 65967

Offset: 0

Seiichi Manyama, May 16 2023

nonn

In general, for m>=1, if g.f. = Sum_{k>=0} x^((m+1)*k) / Product_{j=1..m*k} (1 - x^j), then a(n) ~ Gamma(1/m) * Pi^(1/m) * exp(Pi*sqrt(2*n/3)) / (m^2 * 2^((4*m+1)/(2*m)) * 3^((m+1)/(2*m)) * n^(1 + 1/(2*m))). - Vaclav Kotesovec, Jun 19 2025

			a(8) = 2 counts these partitions:  521, 5111.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Seiichi Manyama)

Cf. A002865, A238479, A363066, A363067.
Cf. A237827, A363047.
Cf. A035297, A035298.

nmax = 60; CoefficientList[Series[Sum[x^(6*k)/Product[1 - x^j, {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 18 2025 *)
nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(5*k))*(1-x^(5*k-1))*(1-x^(5*k-2))*(1-x^(5*k-3))*(1-x^(5*k-4))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(6*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)

a(n) = sum(k=0, n\6, #partitions(n-6*k, 5*k));

G.f.: Sum_{k>=0} x^(6*k)/Product_{j=1..5*k} (1-x^j).

a(n) ~ Gamma(1/5) * Pi^(1/5) * exp(Pi*sqrt(2*n/3)) / (25 * 2^(21/10) * 3^(3/5) * n^(11/10)). - Vaclav Kotesovec, Jun 19 2025

A237829 Number of partitions of n such that 2*(least part) - 1 = greatest part.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 3, 4, 5, 5, 6, 8, 6, 8, 10, 10, 10, 15, 12, 14, 17, 18, 20, 23, 21, 26, 29, 30, 31, 39, 38, 42, 46, 49, 52, 61, 60, 68, 74, 77, 83, 94, 95, 104, 112, 122, 128, 143, 144, 159, 172, 181, 192, 212, 219, 237, 253, 271, 285

Offset: 1

Clark Kimberling, Feb 16 2014

			a(8) = 3 counts these partitions:  53, 332, 11111111.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A118096, A237828.

Cf. A237757, A237825, A237826, A237827, A000041.

z = 64; q[n_] := q[n] = IntegerPartitions[n];
Table[Count[q[n], p_ /; 3 Min[p] == Max[p]], {n, z}]     (* A237825*)
Table[Count[q[n], p_ /; 4 Min[p] == Max[p]], {n, z}]     (* A237826 *)
Table[Count[q[n], p_ /; 5 Min[p] == Max[p]], {n, z}]     (* A237827 *)
Table[Count[q[n], p_ /; 2 Min[p] + 1 == Max[p]], {n, z}] (* A237828 *)
Table[Count[q[n], p_ /; 2 Min[p] - 1 == Max[p]], {n, z}] (* A237829 *)
(* Second program: *)
kmax = 64;
Sum[x^(3k-1)/Product[1-x^j, {j, k, 2k-1}], {k, 1, kmax}]/x+1+O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, May 30 2024, after Seiichi Manyama *)
nmax = 100; p = 1; s = x; Do[p = Simplify[p*(1 - x^(2*k - 1))*(1 - x^(2*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(3*k - 1)/(1 - x^k)*(1 - x^(2*k))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 18 2025 *)

my(N=70, x='x+O('x^N)); Vec(x+sum(k=1, N, x^(3*k-1)/prod(j=k, 2*k-1, 1-x^j))) \\ Seiichi Manyama, May 17 2023

G.f.: x + Sum_{k>=1} x^(3*k-1)/Product_{j=k..2*k-1} (1-x^j). - Seiichi Manyama, May 17 2023

a(n) ~ exp(Pi*sqrt(2*n/15)) / (sqrt(2)* 5^(1/4) * phi^(3/2) * sqrt(n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 20 2025

A363077 Number of partitions of n such that 5*(least part) + 1 = greatest part.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 14, 21, 27, 37, 46, 63, 75, 97, 119, 149, 178, 222, 260, 317, 373, 447, 520, 620, 713, 839, 965, 1123, 1282, 1488, 1687, 1939, 2196, 2508, 2826, 3220, 3610, 4087, 4578, 5157, 5755, 6472, 7199, 8060, 8953, 9991, 11069, 12330, 13625, 15134, 16708, 18508

Offset: 1

Seiichi Manyama, May 17 2023

nonn

Cf. A049820, A237828, A363075, A363076.

Cf. A237827.

nmax = 100; p = 1; s = 0; Do[p = Simplify[p*(1 - x^(5*k - 4))*(1 - x^(5*k - 3))*(1 - x^(5*k - 2))*(1 - x^(5*k - 1))*(1 - x^(5*k))/(1 - x^k)]; p = Normal[p + O[x]^(nmax + 1)]; s += x^(6*k + 1)/(1 - x^k)/(1 - x^(5*k + 1))/p;, {k, 1, nmax}]; Rest[CoefficientList[Series[s, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 19 2025 *)

my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k+1)/prod(j=k, 5*k+1, 1-x^j))))

G.f.: Sum_{k>=1} x^(6*k+1)/Product_{j=k..5*k+1} (1-x^j).

A361459 Number of partitions p of n such that 5*min(p) is a part of p.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 23, 31, 44, 58, 82, 105, 142, 185, 244, 312, 409, 516, 664, 837, 1063, 1328, 1674, 2074, 2588, 3194, 3952, 4847, 5964, 7270, 8884, 10786, 13104, 15832, 19147, 23027, 27709, 33203, 39776, 47476, 56661, 67382, 80108, 94960, 112494, 132919, 156965

Offset: 1

Seiichi Manyama, May 17 2023

nonn

From Vaclav Kotesovec, Jun 19 2025: (Start)
In general, for m>1, if g.f. = Sum_{k>=0} x^(m*k) / Product_{j>=k} (1 - x^j), then A000041(n) - a(n) ~ Pi * (m-1) * exp(Pi*sqrt(2*n/3)) / (3*2^(5/2)*n^(3/2)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)) * (1 - (sqrt(3/2)/Pi + (m - 23/24)*Pi / sqrt(6)) / sqrt(n)). (End)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A117989, A238589, A238590, A238591.

Cf. A237827, A363068.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-6*i, i), i=1..n/6):
seq(a(n), n=1..60);  # Alois P. Heinz, May 17 2023

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - 6 i, i], {i, 1, n/6}];
Array[a, 60] (* Jean-François Alcover, May 30 2024, after Alois P. Heinz *)

my(N=60, x='x+O('x^N)); concat([0, 0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k)/prod(j=k, N, 1-x^j))))

G.f.: Sum_{k>=1} x^(6*k)/Product_{j>=k} (1-x^j).
From Vaclav Kotesovec, Jun 19 2025: (Start)
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 121*Pi/(24*sqrt(6))) / sqrt(n)).
A000041(n) - a(n) ~ 5 * Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). (End)

cp's OEIS Frontend

A118096 Number of partitions of n such that the largest part is twice the smallest part.

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A237828 Number of partitions of n such that 2*(least part) + 1 = greatest part.

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A237825 Number of partitions of n such that 3*(least part) = greatest part.

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A237826 Number of partitions of n such that 4*(least part) = greatest part.

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A363068 Number of partitions p of n such that (1/5)*max(p) is a part of p.

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A237829 Number of partitions of n such that 2*(least part) - 1 = greatest part.

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A363077 Number of partitions of n such that 5*(least part) + 1 = greatest part.

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