A350889 -id:A350889 - cp's OEIS frontend

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).

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).

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).

Original entry on oeis.org

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).

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

A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 20, 22, 26, 30, 35, 40, 48, 55, 65, 76, 90, 105, 126, 147, 175, 206, 244, 286, 339, 396, 467, 545, 638, 741, 865, 1000, 1160, 1337, 1543, 1770, 2035, 2325, 2660, 3029, 3451, 3916, 4447, 5029, 5691, 6419, 7242, 8146, 9167, 10286, 11546, 12930, 14481, 16185

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

			For n=7 there are a(7)=3 such partitions: [1,2,2,2], [1,1,2,3] and [1,1,1,4]. - _R. J. Mathar_, Jun 20 2022

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

Column 4 of A350889.

Cf. A168657.

CoefficientList[Series[Sum[x^(4k^2)/Product[1-x^j,{j,4k-1}],{k,63}],{x,0,63}],x] (* Stefano Spezia, Jan 22 2022 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, 4*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(4*k^2)/Product_{j=1..4*k-1} (1-x^j).

a(n) ~ c * exp(Pi*sqrt(2*n/5)) / n^(3/4), where c = (3 - sqrt(5))^(1/4) / (8*sqrt(5)) = 0.05226232058... - Vaclav Kotesovec, Jan 25 2022, updated Oct 13 2024

A350897 Number of partitions of n such that 5*(smallest part) = (number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 55, 65, 74, 87, 99, 115, 131, 151, 172, 199, 226, 260, 298, 343, 393, 454, 522, 603, 696, 804, 929, 1076, 1243, 1438, 1664, 1924, 2222, 2567, 2961, 3413, 3931, 4520, 5193, 5959, 6827, 7811, 8928, 10186, 11607, 13208, 15008, 17028, 19297

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m. - Vaclav Kotesovec, Oct 14 2024

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

Column 5 of A350889.

Cf. A168657.

CoefficientList[Series[Sum[x^(5k^2)/Product[1-x^j,{j,5k-1}],{k,62}],{x,0,62}],x] (* Stefano Spezia, Jan 22 2022 *)

my(N=66, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, 5*k-1, 1-x^j))))

G.f.: Sum_{k>=1} x^(5*k^2)/Product_{j=1..5*k-1} (1-x^j).
a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 0.42067169741517... and c = 0.04778365700734... - Vaclav Kotesovec, Jan 26 2022
a(n) ~ r^2 * (5*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((5*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(5*Pi*(5 - 3*r^2)) * n^(3/4)), where r = 0.808730600479392... is the real root of the equation r^2 = 1 - r^5. - Vaclav Kotesovec, Oct 14 2024

A377075 G.f.: Sum_{k>=0} x^(8k^2) / Product_{j=1..8k-1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 21, 28, 38, 49, 65, 82, 105, 131, 164, 201, 248, 300, 364, 436, 522, 618, 734, 861, 1011, 1178, 1372, 1586, 1835, 2108, 2422, 2768, 3162, 3595, 4088, 4627, 5237, 5907, 6660, 7485, 8414, 9429, 10568, 11817, 13213

Offset: 0

Vaclav Kotesovec, Oct 15 2024

nonn

In general, for m >= 1, if g.f.= Sum_{k>=1} x^(m*k^2)/Product_{j=1..m*k-1} (1-x^j), then a(n) ~ r^2 * (m*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((m*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*m*(m - (m-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^m.

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

Column 8 of A350889.

Cf. A376658.

nmax = 100; CoefficientList[Series[Sum[x^(8*k^2)/Product[1-x^j, {j, 1, 8*k-1}], {k, 1, Sqrt[nmax/8]}], {x, 0, nmax}], x]

Limit_{n->oo} a(n)^(1/sqrt(n)) = A376658.

a(n) ~ r^2 * (8*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((8*log(r)^2 + polylog(2, r^2))*n)) / (8*sqrt(Pi*(4 - 3*r^2)) * n^(3/4)), where r = 0.8511709340670154789... is the positive real root of the equation r^2 = 1 - r^8.

A377076 G.f.: Sum_{k>=0} x^(6k^2) / Product_{j=1..6k-1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 142, 165, 194, 224, 260, 298, 344, 392, 449, 510, 582, 659, 750, 847, 962, 1087, 1233, 1393, 1581, 1787, 2029, 2297, 2610, 2958, 3365, 3819, 4348, 4942, 5630, 6404, 7302, 8310, 9475, 10787

Offset: 0

Vaclav Kotesovec, Oct 15 2024

nonn

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

Column 6 of A350889.

nmax = 100; CoefficientList[Series[Sum[x^(6*k^2)/Product[1-x^j, {j, 1, 6*k-1}], {k, 1, Sqrt[nmax/6]}], {x, 0, nmax}], x]

a(n) ~ r^2 * (6*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((6*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(12*Pi*(3 - 2*r^2)) * n^(3/4)), where r = sqrt(((9 + sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9 + sqrt(93))))^(1/3)) = 0.82603135765418... is the positive real root of the equation r^2 = 1 - r^6.

A377077 G.f.: Sum_{k>=0} x^(7k^2) / Product_{j=1..7k-1} (1 - x^j).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 20, 26, 35, 44, 58, 71, 90, 110, 136, 163, 199, 235, 282, 332, 392, 456, 535, 617, 716, 822, 946, 1079, 1236, 1402, 1596, 1806, 2046, 2306, 2606, 2929, 3299, 3704, 4163, 4667, 5241, 5870, 6585, 7378, 8273, 9268, 10397

Offset: 0

Vaclav Kotesovec, Oct 15 2024

nonn

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

Column 7 of A350889.

nmax = 100; CoefficientList[Series[Sum[x^(7*k^2)/Product[1-x^j, {j, 1, 7*k-1}], {k, 1, Sqrt[nmax/7]}], {x, 0, nmax}], x]

a(n) ~ r^2 * (7*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((7*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(7*Pi*(7 - 5*r^2)) * n^(3/4)), where r = 0.839833147032421662... is the positive real root of the equation r^2 = 1 - r^7.

cp's OEIS Frontend

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

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A350896 Number of partitions of n such that 4*(smallest part) = (number of parts).

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A350897 Number of partitions of n such that 5*(smallest part) = (number of parts).

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A377075 G.f.: Sum_{k>=0} x^(8*k^2) / Product_{j=1..8*k-1} (1 - x^j).

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A377076 G.f.: Sum_{k>=0} x^(6*k^2) / Product_{j=1..6*k-1} (1 - x^j).

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A377077 G.f.: Sum_{k>=0} x^(7*k^2) / Product_{j=1..7*k-1} (1 - x^j).

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A377075 G.f.: Sum_{k>=0} x^(8k^2) / Product_{j=1..8k-1} (1 - x^j).

A377076 G.f.: Sum_{k>=0} x^(6k^2) / Product_{j=1..6k-1} (1 - x^j).

A377077 G.f.: Sum_{k>=0} x^(7k^2) / Product_{j=1..7k-1} (1 - x^j).