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).

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

Author

Seiichi Manyama, Jan 21 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			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;
		

Links

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

Crossrefs

Row sums give A168657.

Column k=1..8 give A006141, A237757, A350892, A350896, A350897, A377076, A377077, A377075.

Cf. A350879, A350890.

Programs

PARI

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);

Ruby

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)

Formula

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