A350890 -id:A350890 - 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).

A350889 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*(smallest part) = (number of parts).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 2, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 2, 2, 4, 5, 5, 3, 2, 1, 1, 2, 3, 4, 7, 6, 5, 3, 2, 1, 1, 3, 4, 5, 8, 9, 7, 5, 3, 2, 1, 1, 3, 5, 6, 10, 11, 10, 7, 5, 3, 2, 1, 1, 4, 6, 7, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 4, 8, 8, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Seiichi Manyama, Jan 21 2022

Column k is asymptotic to r^2 * (k*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((k*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(Pi*k*(k - (k-2)*r^2)) * n^(3/4)), where r is the positive real root of the equation r^2 = 1 - r^k. - Vaclav Kotesovec, Oct 14 2024

			Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  1, 1, 1, 1;
  1, 1, 2, 1, 1;
  1, 1, 2, 2, 1, 1;
  1, 1, 3, 3, 2, 1, 1;
  1, 2, 3, 4, 3, 2, 1, 1;
  2, 2, 4, 5, 5, 3, 2, 1, 1;
  2, 3, 4, 7, 6, 5, 3, 2, 1, 1;
  3, 4, 5, 8, 9, 7, 5, 3, 2, 1, 1;

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

Row sums give A168657.
Column k=1..8 give A006141, A237757, A350892, A350896, A350897, A377076, A377077, A377075.
Cf. A350879, A350890.

T(n, k) = polcoef(sum(i=1, sqrtint(n\k), x^(k*i^2)/prod(j=1, k*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 i * ary[-1] == ary.size
    }
  }
  a
end
def A350889(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350889(14)

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

A350893 Number of partitions of n such that (smallest part) = 2*(number of parts).

Original entry on oeis.org

0, 1, 0, 0, 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, 10, 10, 12, 13, 15, 16, 19, 20, 23, 25, 28, 30, 34, 36, 40, 43, 47, 50, 56, 59, 65, 70, 77, 82, 91, 97, 107, 115, 126, 135, 149, 159, 174, 187, 204, 218, 238, 254, 276, 295, 320, 341, 370, 394, 426, 455, 491, 523, 565

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 2 of A350890.

Cf. A168656, A237753, A237757.

nmax = 100; Rest[CoefficientList[1 + Series[Sum[x^(2*j^2)*(1 - x^j)/Product[1 - x^i, {i, 1, j}], {j, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 21 2022 *)

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

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(2*log(alfa)^2 + polylog(2, 1 - alfa)))) * (2*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(4 - 3*alfa) * n^(3/4)), where alfa = 0.72449195900051561158837228218703656578649448135... is positive real root of the equation alfa^4 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 21 2022

A350898 Number of partitions of n such that (smallest part) = 4*(number of parts).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 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, 15, 15, 17, 17, 19, 20, 22, 23, 26, 27, 30, 32, 35, 37, 41, 43, 47, 50, 54, 57, 62, 65, 70, 74, 79, 83, 89, 93, 99, 104

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 4 of A350890.

Cf. A168656.

my(N=99, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, sqrtint(N\4), x^(4*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(4*log(alfa)^2 + polylog(2, 1 - alfa)))) * (4*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(8 - 7*alfa) * n^(3/4)), where alfa = 0.8116523200278026483934188589034567041719182934245... is positive real root of the equation alfa^8 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 5, 3, 1, 1, 0, 7, 3, 2, 1, 1, 0, 11, 6, 4, 2, 1, 1, 0, 15, 7, 5, 3, 2, 1, 1, 0, 22, 12, 7, 6, 3, 2, 1, 1, 0, 30, 14, 11, 7, 5, 3, 2, 1, 1, 0, 42, 22, 14, 11, 8, 5, 3, 2, 1, 1, 0, 56, 27, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 77, 40, 27, 21, 15, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Seiichi Manyama, May 14 2023

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  1,  1;
  0,  5,  3,  1, 1;
  0,  7,  3,  2, 1, 1;
  0, 11,  6,  4, 2, 1, 1;
  0, 15,  7,  5, 3, 2, 1, 1;
  0, 22, 12,  7, 6, 3, 2, 1, 1;
  0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Row sums give A323433.
Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
T(2n,n) gives A052810.
Cf. A008284, A072233, A350890.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
    (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 14 2023

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));

For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).

A350899 Number of partitions of n such that (smallest part) = 5*(number of parts).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 15, 15, 16, 16, 17, 17, 18, 18, 19, 20, 21, 22, 24, 25, 27, 29, 31, 33, 36, 38, 41, 44, 47

Offset: 1

Seiichi Manyama, Jan 21 2022

nonn

Column 5 of A350890.

Cf. A168656.

my(N=99, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(sum(k=1, sqrtint(N\5), x^(5*k^2)/prod(j=1, k-1, 1-x^j))))

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

a(n) ~ (1 - alfa) * exp(2*sqrt(n*(5*log(alfa)^2 + polylog(2, 1 - alfa)))) * (5*log(alfa)^2 + polylog(2, 1 - alfa))^(1/4) / (2*sqrt(Pi) * sqrt(10 - 9*alfa) * n^(3/4)), where alfa = 0.8350790427235590476091499923248865165628469558282... is positive real root of the equation alfa^10 + alfa - 1 = 0. - Vaclav Kotesovec, Jan 22 2022

A377080 G.f.: Sum_{k>=1} x^(2k^2) Product_{j=1..k} (1 + x^j).

Original entry on oeis.org

0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 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, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4

Offset: 0

Vaclav Kotesovec, Oct 15 2024

nonn

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

Cf. A306734, A350890, A350893, A377081.

Cf. A230152.

nmax = 150; CoefficientList[Series[Sum[x^(2*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/2]}], {x, 0, nmax}], x]

a(n) ~ (1+r) * exp(sqrt((40*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((4 + 5*r)*n)), where r = A230152 = 0.856674883854502874852324... is the real root of the equation r^4*(1+r) = 1.

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

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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A350889 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*(smallest 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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A350893 Number of partitions of n such that (smallest part) = 2*(number of parts).

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A350898 Number of partitions of n such that (smallest part) = 4*(number of parts).

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A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

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A350899 Number of partitions of n such that (smallest part) = 5*(number of parts).

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A377080 G.f.: Sum_{k>=1} x^(2*k^2) * Product_{j=1..k} (1 + x^j).

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A377080 G.f.: Sum_{k>=1} x^(2k^2) Product_{j=1..k} (1 + x^j).

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