A237756 -id:A237756 - cp's OEIS frontend

A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts).

1, 2, 2, 4, 6, 9, 12, 18, 24, 34, 46, 63, 83, 111, 144, 190, 245, 318, 405, 520, 657, 833, 1045, 1312, 1634, 2036, 2517, 3114, 3829, 4705, 5751, 7027, 8544, 10381, 12564, 15190, 18301, 22026, 26425, 31669, 37849, 45180, 53796, 63983, 75923, 89987, 106435

Offset: 1

Clark Kimberling, Feb 13 2014

Also, the number of partitions of n such that (greatest part) <= 2*(number of parts); hence, the number of partitions of n such that (rank + greatest part) >= 0.

			a(6) = 9 counts all of the 11 partitions of 6 except these:  21111, 111111.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Cf. A064173, A237751-A237755, A237756, A237757, A000041.

z = 50; Table[Count[IntegerPartitions[n], p_ /; 2 Max[p] >= Length[p]], {n, z}]

{a(n) = my(A); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(2*m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015

a(n) = A000041(n) - A237751(n).

G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(2*n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015

A350879 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of partitions of n such that k*(greatest part) = (number of parts).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 1, 4, 1, 1, 1, 0, 0, 0, 0, 1, 4, 2, 1, 1, 0, 0, 0, 0, 0, 1, 6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 1, 7, 4, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 11, 5, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 11, 7, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Seiichi Manyama, Jan 21 2022

T(n,k) is the number of partitions of n such that (greatest part) = k*(number of parts).

Column k > 1 is asymptotic to k! * Pi^k * exp(sqrt(2*Pi*n/3)) / (2^((k+4)/2) * 3^((k+1)/2) * n^((k+2)/2)). Equivalently, for fixed k > 1, T(n,k) ~ k! * Pi^k * A000041(n) / (6^(k/2) * n^(k/2)). - Vaclav Kotesovec, Oct 17 2024

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  1, 0, 0, 1;
  1, 1, 0, 0, 1;
  1, 1, 0, 0, 0, 1;
  3, 1, 1, 0, 0, 0, 1;
  2, 2, 1, 0, 0, 0, 0, 1;
  4, 1, 1, 1, 0, 0, 0, 0, 1;
  4, 2, 1, 1, 0, 0, 0, 0, 0, 1;
  6, 3, 2, 1, 1, 0, 0, 0, 0, 0, 1;

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50).

Row sums give A168659.
Column k=1..7 give A047993, A237753, A237756, A348163, A348164, A377107, A377108.
Cf. A063995, A350889, A350890.

T(n, k) = polcoef(sum(i=1, (n+1)\(k+1), x^((k+1)*i-1)*prod(j=1, i-1, (1-x^(k*i+j-1))/(1-x^j+x*O(x^n)))), n);

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A(n)
  a = Array.new(n, 0)
  partition(n, 1, n).each{|ary|
    (1..n).each{|i|
      a[i - 1] += 1 if i * ary[0] == ary.size
    }
  }
  a
end
def A350879(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350879(14)

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

A350894 Number of partitions of n such that (smallest part) = 3*(number of parts).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 13, 14, 15, 17, 18, 20, 22, 24, 26, 29, 31, 34, 37, 40, 43, 47, 50, 54, 58, 62, 66, 71, 75, 80, 85, 90, 95, 102, 107, 114, 121, 129, 136, 146, 154, 165, 175, 187, 198, 213

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

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

Column 3 of A350890.

Cf. A168656, A237756.

my(N=99, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(3*log(alfa)^2 + polylog(2, 1 - alfa)))) * (3*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(6 - 5*alfa) * n^(3/4)), where alfa = 0.7780895986786010978806823096592944458720784440255... is positive real root of the equation alfa^6 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 12, 15, 18, 22, 27, 33, 40, 48, 58, 69, 82, 98, 115, 135, 158, 184, 214, 248, 286, 330, 379, 435, 497, 569, 648, 739, 840, 955, 1082, 1228, 1388, 1572, 1775, 2005, 2259, 2549, 2867, 3228, 3626, 4076, 4571, 5131, 5745, 6438, 7199, 8053, 8992, 10045, 11199

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..20000

Column 3 of A350889.

Cf. A168657, A237756, A350894.

CoefficientList[Series[Sum[x^(3k^2)/Product[1-x^j,{j,3k-1}],{k,64}],{x,0,64}],x] (* Stefano Spezia, Jan 22 2022 *)
Table[Count[IntegerPartitions[n],?(3#[[-1]]==Length[#]&)],{n,70}] (* _Harvey P. Dale, Jul 13 2023 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, sqrtint(N\3), x^(3*k^2)/prod(j=1, 3*k-1, 1-x^j))))

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

a(n) ~ c * exp(2*sqrt((5*log(A075778)^2 + 2*polylog(2, 1 - A075778))*n)) / n^(3/4), where c = (3*log(A075778)^2 + polylog(2, A075778^2))^(1/4) / (2*sqrt(3*Pi*(1 + A075778)*(2 + 3*A075778))) = 0.0582980106266835787... - Vaclav Kotesovec, Jan 24 2022, updated Oct 14 2024

A347867 Number of partitions of n such that 3*(greatest part) >= (number of parts).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 14, 20, 27, 38, 51, 70, 92, 123, 162, 212, 274, 355, 453, 579, 733, 928, 1165, 1463, 1822, 2269, 2808, 3470, 4266, 5241, 6407, 7823, 9514, 11554, 13983, 16900, 20359, 24494, 29386, 35205, 42069, 50206, 59773, 71069, 84322, 99913, 118157, 139556, 164528, 193734

Offset: 1

Seiichi Manyama, Jan 25 2022

nonn

Also, the number of partitions of n such that (greatest part) <= 3*(number of parts).

Cf. A064174, A237755, A347868, A347869.

Cf. A237756.

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(3*k+j-1))/(1-x^j))))

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

cp's OEIS Frontend

A237755 Number of partitions of n such that 2*(greatest part) >= (number of parts).

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A350879 Triangle T(n,k), n >= 1, 1 <= k <= n, read by rows, where T(n,k) is the number of partitions of n such that k*(greatest part) = (number of parts).

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A350894 Number of partitions of n such that (smallest part) = 3*(number of parts).

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A350892 Number of partitions of n such that 3*(smallest part) = (number of parts).

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A347867 Number of partitions of n such that 3*(greatest part) >= (number of parts).

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