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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).
10
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.
Sequence in context: A279281 A342465 A124749 * A331416 A127844 A017877
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved