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%I #25 May 06 2023 11:13:12
%S 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,
%T 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,
%U 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
%N 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).
%H Andrew Howroyd, <a href="/A350889/b350889.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50).
%F G.f. of column k: Sum_{i>=1} x^(k*i^2)/Product_{j=1..k*i-1} (1-x^j).
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 1, 1, 1, 1;
%e 1, 1, 2, 1, 1;
%e 1, 1, 2, 2, 1, 1;
%e 1, 1, 3, 3, 2, 1, 1;
%e 1, 2, 3, 4, 3, 2, 1, 1;
%e 2, 2, 4, 5, 5, 3, 2, 1, 1;
%e 2, 3, 4, 7, 6, 5, 3, 2, 1, 1;
%e 3, 4, 5, 8, 9, 7, 5, 3, 2, 1, 1;
%o (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);
%o (Ruby)
%o def partition(n, min, max)
%o return [[]] if n == 0
%o [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o end
%o def A(n)
%o a = Array.new(n, 0)
%o partition(n, 1, n).each{|ary|
%o (1..n).each{|i|
%o a[i - 1] += 1 if i * ary[-1] == ary.size
%o }
%o }
%o a
%o end
%o def A350889(n)
%o (1..n).map{|i| A(i)}.flatten
%o end
%o p A350889(14)
%Y Row sums give A168657.
%Y Column k=1..5 give A006141, A237757, A350892, A350896, A350897.
%Y Cf. A350879, A350890.
%K nonn,tabl
%O 1,13
%A _Seiichi Manyama_, Jan 21 2022
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