OFFSET
1,22
COMMENTS
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
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+1)*i-1) * Product_{j=1..i-1} (1-x^(k*i+j-1))/(1-x^j).
EXAMPLE
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;
PROG
(PARI) 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);
(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[0] == ary.size
}
}
a
end
def A350879(n)
(1..n).map{|i| A(i)}.flatten
end
p A350879(14)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved