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

1,37

Comments

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

Links

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

Formula

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

Example

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;

Prog

(PARI) 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);
(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 ary[-1] == i * ary.size
    }
  }
  a
end
def A350890(n)
  (1..n).map{|i| A(i)}.flatten
end
p A350890(14)

Crossrefs

Row sums give A168656.

Column k=1..5 give A006141, A350893, A350894, A350898, A350899.

Cf. A350879, A350889.

Sequence in context: A279281 A342465 A124749 * A331416 A127844 A017877

Adjacent sequences: A350887 A350888 A350889 * A350891 A350892 A350893

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 21 2022

Status

approved