A238479 -id:A238479 - cp's OEIS frontend

A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 51, 64, 81, 102, 128, 159, 198, 245, 304, 374, 460, 563, 689, 841, 1023, 1242, 1505, 1819, 2195, 2642, 3173, 3804, 4551, 5435, 6477, 7707, 9151, 10850, 12843, 15175, 17902, 21089, 24802, 29132, 34164, 40012, 46796, 54663, 63766

Offset: 0

Seiichi Manyama, May 16 2023

nonn

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A002865, A238479, A363066, A363068.
Cf. A237826, A363046.
Cf. A035296.

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

a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));

G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).

a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Jun 19 2025

A238625 Number of partitions p of n such that 1 + (1/2)*max(p) is a part of p.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 19, 24, 31, 41, 51, 65, 84, 105, 132, 167, 207, 257, 321, 395, 486, 599, 731, 892, 1089, 1319, 1597, 1933, 2327, 2798, 3361, 4021, 4805, 5736, 6825, 8109, 9625, 11393, 13469, 15905, 18738, 22049, 25915, 30401, 35620

Offset: 1

Clark Kimberling, Mar 02 2014

			a(6) counts these partitions:  222, 2211, 21111.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A238479, A238624.

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, 1 + Max[p]/2]], {n, 50}]
p[n_, m_] := If[m > n, 0, If[n == m, 1, p[n, m] = Sum[p[n - m, j], {j, m}]]]; a[1] = 0; a[n_] := 1 + Sum[p[n-k-1, 2*k], {k, n/2}]; Array[a,100] (* Giovanni Resta, Mar 07 2014 *)

A308151 Triangular array: each row partitions the partitions of n into n parts; of which the k-th part is the number of partitions having stay number k-1; see Comments.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 3, 1, 0, 0, 1, 3, 3, 2, 2, 0, 0, 1, 4, 6, 2, 1, 1, 0, 0, 1, 5, 8, 4, 1, 2, 1, 0, 0, 1, 8, 10, 4, 4, 1, 1, 1, 0, 0, 1, 10, 14, 8, 3, 2, 2, 1, 1, 0, 0, 1, 13, 20, 9, 5, 3, 2, 1, 1, 1, 0, 0, 1, 18, 25, 12, 8, 5, 2

Offset: 1

Clark Kimberling, May 16 2019

The stay number of a partition P is defined as follows. Let U be the ordering of the parts of P in nonincreasing order, and let V be the reverse of U. The stay number of P is the number of numbers whose position in V is the same as in U. (1st column) = A238479. When the rows of the array are read in reverse order, it appears that the limiting sequence is A008483.

			The first 8 rows:
  1
  0   1
  0   1   1
  1   1   0   1
  1   2   1   0   1
  2   3   1   0   0   1
  3   3   2   2   0   0   1
  4   6   2   1   1   0   0   1
  5   8   4   1   2   1   0   0   1
For n = 5, P consists of these partitions:
[5], with reversal [5], thus, 1 stay number
[4,1], with reversal [1,4], thus 0 stay numbers
[3,2], with reversal [2,3], thus 0 stay numbers
[2,2,1], with reversal [1,2,2], thus 1 stay number
[2,1,1,1], with reversal [1,1,1,2], thus 2 stay numbers
[1,1,1,1,1], thus, 5 stay numbers.
As a result, row 5 of the array is 2 3 1 0 0 1

Cf. A000041, A008483, A238479.

Map[BinCounts[#, {0, Last[#] + 1, 1}] &,  Map[Map[Count[#, 0] &, # - Map[Reverse, #] &[IntegerPartitions[#]]] &, Range[0, 35]]]
  (* Peter J. C. Moses, May 14 2019 *)

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A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

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A238625 Number of partitions p of n such that 1 + (1/2)*max(p) is a part of p.

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A308151 Triangular array: each row partitions the partitions of n into n parts; of which the k-th part is the number of partitions having stay number k-1; see Comments.

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