cp's OEIS Frontend

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).

%I A350890 #25 May 06 2023 11:12:45
%S A350890 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,
%T A350890 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,
%U A350890 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
%N 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).
%C A350890 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
%H A350890 Andrew Howroyd, <a href="/A350890/b350890.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50).
%F A350890 G.f. of column k: Sum_{i>=1} x^(k*i^2)/Product_{j=1..i-1} (1-x^j).
%e A350890 Triangle begins:
%e A350890   1;
%e A350890   0, 1;
%e A350890   0, 0, 1;
%e A350890   1, 0, 0, 1;
%e A350890   1, 0, 0, 0, 1;
%e A350890   1, 0, 0, 0, 0, 1;
%e A350890   1, 0, 0, 0, 0, 0, 1;
%e A350890   1, 1, 0, 0, 0, 0, 0, 1;
%e A350890   2, 1, 0, 0, 0, 0, 0, 0, 1;
%e A350890   2, 1, 0, 0, 0, 0, 0, 0, 0, 1;
%e A350890   3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%o A350890 (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);
%o A350890 (Ruby)
%o A350890 def partition(n, min, max)
%o A350890   return [[]] if n == 0
%o A350890   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A350890 end
%o A350890 def A(n)
%o A350890   a = Array.new(n, 0)
%o A350890   partition(n, 1, n).each{|ary|
%o A350890     (1..n).each{|i|
%o A350890       a[i - 1] += 1 if ary[-1] == i * ary.size
%o A350890     }
%o A350890   }
%o A350890   a
%o A350890 end
%o A350890 def A350890(n)
%o A350890   (1..n).map{|i| A(i)}.flatten
%o A350890 end
%o A350890 p A350890(14)
%Y A350890 Row sums give A168656.
%Y A350890 Column k=1..5 give A006141, A350893, A350894, A350898, A350899.
%Y A350890 Cf. A350879, A350889.
%K A350890 nonn,tabl
%O A350890 1,37
%A A350890 _Seiichi Manyama_, Jan 21 2022