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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). 6
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,13
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..k*i-1} (1-x^j).
EXAMPLE
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;
PROG
(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)
CROSSREFS
Row sums give A168657.
Column k=1..5 give A006141, A237757, A350892, A350896, A350897.
Sequence in context: A140356 A119963 A057790 * A224697 A052307 A067059
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)