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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).
11
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
1,22
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
Row sums give A168659.
Sequence in context: A338211 A115717 A339632 * A115718 A361510 A204181
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved