A350899 -id:A350899 - cp's OEIS frontend

A350890 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 (smallest part) = k*(number of parts).

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Seiichi Manyama, Jan 21 2022

Column k is asymptotic to (1 - alfa) * exp(2*sqrt(n*(k*log(alfa)^2 + polylog(2, 1 - alfa)))) * (k*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(alfa + 2*k - 2*alfa*k) * n^(3/4)), where alfa is positive real root of the equation alfa^(2*k) + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022

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

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

Row sums give A168656.
Column k=1..5 give A006141, A350893, A350894, A350898, A350899.
Cf. A350879, A350889.

T(n, k) = polcoef(sum(i=1, sqrtint(n\k), x^(k*i^2)/prod(j=1, i-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 ary[-1] == i * ary.size
    }
  }
  a
end
def A350890(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350890(14)

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

A373070 Number of partitions of n such that (smallest part) >= 5*(number of parts).

Original entry on oeis.org

1, 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, 12, 13, 13, 14, 15, 16, 17, 19, 20, 22, 24, 26, 28, 31, 33, 36, 39, 42, 45, 49, 52, 56, 60, 64, 68, 73, 77, 82, 87, 92, 97, 103, 108, 114, 120, 126, 132, 139, 145, 153, 160, 168, 176, 186, 194

Offset: 0

Seiichi Manyama, May 22 2024

nonn

Cf. A003114, A373067, A373068, A373069.

Cf. A350899, A373076.

my(N=90, x='x+O('x^N)); Vec(sum(k=0, N, x^(5*k^2)/prod(j=1, k, 1-x^j)))

G.f.: Sum_{k>=0} x^(5*k^2)/Product_{j=1..k} (1-x^j).

A373076 Number of partitions of n such that (smallest part) > 5*(number of parts).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 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, 12, 13, 13, 14, 14, 16, 16, 18, 19, 21, 22, 25, 26, 29, 31, 34, 36, 40, 42, 46, 49, 53, 56, 61, 64, 69, 73, 78, 82, 88, 92, 98, 103, 109, 114, 121, 126, 133, 139, 146, 152, 161

Offset: 0

Seiichi Manyama, May 22 2024

nonn

This sequence is different from A350898.

Cf. A003106, A373073, A373074, A373075.

Cf. A350899, A373070.

Join[{1},Table[Count[IntegerPartitions[n],?(#[[-1]]>5Length[#]&)],{n,90}]] (* _Harvey P. Dale, May 24 2025 *)

my(N=90, x='x+O('x^N)); Vec(sum(k=0, N, x^(5*k^2+k)/prod(j=1, k, 1-x^j)))

G.f.: Sum_{k>=0} x^(5*k^2+k)/Product_{j=1..k} (1-x^j).

A348164 Number of partitions of n such that 5*(greatest part) = (number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 5, 5, 7, 8, 10, 11, 15, 16, 20, 22, 26, 28, 35, 38, 46, 52, 62, 70, 85, 95, 112, 127, 148, 166, 195, 219, 254, 288, 332, 375, 435, 489, 562, 635, 726, 817, 936, 1051, 1198, 1348, 1531, 1721, 1957, 2196, 2489

Offset: 1

Seiichi Manyama, Jan 25 2022

nonn

Also, the number of partitions of n such that (greatest part) = 5*(number of parts).

			a(19) = 3 counts these partitions:
[3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1],
[2, 2, 2, 2, 2, 2, 2, 2, 2, 1].

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

Column 5 of A350879.

Cf. A350897, A350899.

nmax = 100; Rest[CoefficientList[Series[Sum[x^(6*k-1) * Product[(1 - x^(5*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/6 + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 15 2024 *)
nmax = 100; p = x^4; s = x^4; Do[p = Normal[Series[p*x^6*(1 - x^(6*k - 1))*(1 - x^(6*k))*(1 - x^(6*k + 1))*(1 - x^(6*k + 2))*(1 - x^(6*k + 3))*(1 - x^(6*k + 4))/((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)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/6 + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, N, x^(6*k-1)*prod(j=1, k-1, (1-x^(5*k+j-1))/(1-x^j)))))

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

a(n) ~ 5 * Pi^5 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(3/2) * n^(7/2)). - Vaclav Kotesovec, Oct 17 2024

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A350890 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 (smallest part) = k*(number of parts).

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A373070 Number of partitions of n such that (smallest part) >= 5*(number of parts).

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A373076 Number of partitions of n such that (smallest part) > 5*(number of parts).

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A348164 Number of partitions of n such that 5*(greatest part) = (number of parts).

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