A350894 -id:A350894 - 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).

A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 6, 7, 10, 10, 13, 14, 19, 21, 27, 31, 40, 45, 55, 64, 79, 91, 111, 127, 154, 177, 211, 243, 290, 333, 394, 455, 538, 618, 726, 834, 977, 1121, 1304, 1495, 1738, 1989, 2302, 2633, 3041, 3473, 3999, 4562, 5241

Offset: 1

Clark Kimberling, Feb 13 2014

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

			a(15) = 4 counts these partitions: [12,1,1,1], [9,5,1], [9,4,2], [9,3,3].

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Seiichi Manyama)

Column 3 of A350879.

Cf. A064173, A237753, A350892, A350894.

z = 50; Table[Count[IntegerPartitions[n], p_ /; Max[p] = = 3 Length[p]], {n, z}]
(* or *)
nmax = 100; Rest[CoefficientList[Series[Sum[x^(4*k-1) * Product[(1 - x^(3*k+j-1)) / (1 - x^j), {j, 1, k-1}], {k, 1, nmax/4 + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 15 2024 *)
nmax = 100; p = x^2; s = x^2; Do[p = Normal[Series[p*x^4*(1 - x^(4*k - 1))*(1 - x^(4*k))*(1 - x^(4*k + 1))*(1 - x^(4*k + 2))/((1 - x^(3*k + 2))*(1 - x^(3*k + 1))*(1 - x^(3*k))*(1 - x^k)), {x, 0, nmax}]]; s += p;, {k, 1, nmax/4 + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(4*k-1)*prod(j=1, k-1, (1-x^(3*k+j-1))/(1-x^j))))) \\ Seiichi Manyama, Jan 24 2022

G.f.: Sum_{k>=1} x^(4*k-1) * Product_{j=1..k-1} (1-x^(3*k+j-1))/(1-x^j). - Seiichi Manyama, Jan 24 2022

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

A373068 Number of partitions of n such that (smallest part) >= 3*(number of parts).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 12, 14, 15, 17, 19, 21, 23, 26, 28, 31, 34, 37, 40, 44, 47, 51, 55, 59, 63, 69, 73, 79, 85, 92, 98, 107, 114, 124, 133, 144, 154, 168, 179, 194, 208, 225, 240, 260, 277, 299, 319, 343, 365, 393, 417, 447, 476

Offset: 0

Seiichi Manyama, May 22 2024

nonn

Cf. A003114, A373067, A373069, A373070.

Cf. A350894, A373074.

my(N=80, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k^2)/prod(j=1, k, 1-x^j)))

G.f.: Sum_{k>=0} x^(3*k^2)/Product_{j=1..k} (1-x^j).

A373074 Number of partitions of n such that (smallest part) > 3*(number of parts).

Original entry on oeis.org

1, 0, 0, 0, 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, 11, 11, 13, 14, 16, 17, 20, 21, 24, 26, 29, 31, 35, 37, 41, 44, 48, 51, 56, 59, 64, 68, 74, 78, 85, 90, 98, 104, 113, 120, 131, 139, 151, 161, 175, 186, 202, 215, 233, 248, 268, 285, 308, 327, 352, 374, 402, 426, 457

Offset: 0

Seiichi Manyama, May 22 2024

nonn

Cf. A003106, A373073, A373075, A373076.

Cf. A350894, A373068.

Join[{1},Table[Count[IntegerPartitions[n],?(#[[-1]]>3*Length[#]&)],{n,80}]] (* _Harvey P. Dale, Aug 15 2024 *)

my(N=80, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k^2+k)/prod(j=1, k, 1-x^j)))

G.f.: Sum_{k>=0} x^(3*k^2+k)/Product_{j=1..k} (1-x^j).

A373075 Number of partitions of n such that (smallest part) > 4*(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, 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, 101, 106, 113, 119, 126, 133, 142, 149, 159, 168, 179, 189, 202

Offset: 0

Seiichi Manyama, May 22 2024

nonn

This sequence is different from A350894.

Cf. A003106, A373073, A373074, A373076.

Cf. A350898, A373069.

my(N=80, x='x+O('x^N)); Vec(sum(k=0, N, x^(4*k^2+k)/prod(j=1, k, 1-x^j)))

G.f.: Sum_{k>=0} x^(4*k^2+k)/Product_{j=1..k} (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

A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Oct 15 2024

In general, if m > 0 and g.f. = Sum_{k>=1} x^(m*k^2) * Product_{j=1..k} (1 + x^j), then a(n) ~ (1+r) * exp(sqrt((4*m*(2*m+1)*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((r + 2*m*(1+r))*n)), where r is the smallest positive real root of the equation r^(2*m)*(1+r) = 1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A306734 (m=1), A377080 (m=2).

Cf. A230154, A350890, A350894.

nmax = 200; CoefficientList[Series[Sum[x^(3*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/3]}], {x, 0, nmax}], x]

a(n) ~ (1+r) * exp(sqrt((84*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((6 + 7*r)*n)), where r = A230154 = 0.898653712628699293260875722... is the real root of the equation r^6*(1+r) = 1.

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).

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A237756 Number of partitions of n such that 3*(greatest part) = (number of parts).

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A373068 Number of partitions of n such that (smallest part) >= 3*(number of parts).

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A373074 Number of partitions of n such that (smallest part) > 3*(number of parts).

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A373075 Number of partitions of n such that (smallest part) > 4*(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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A377081 G.f.: Sum_{k>=1} x^(3*k^2) * Product_{j=1..k} (1 + x^j).

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A377081 G.f.: Sum_{k>=1} x^(3k^2) Product_{j=1..k} (1 + x^j).