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
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jan 21 2022
STATUS
approved