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A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.

%I A194543 #51 Oct 04 2018 19:45:08
%S A194543 1,1,1,2,1,1,3,2,1,1,5,2,2,1,1,7,3,2,2,1,1,11,4,3,2,2,1,1,15,5,3,3,2,
%T A194543 2,1,1,22,6,4,3,3,2,2,1,1,30,8,5,4,3,3,2,2,1,1,42,10,6,4,4,3,3,2,2,1,
%U A194543 1,56,12,7,5,4,4,3,3,2,2,1,1,77,15,9,6,5,4,4,3,3,2,2,1,1
%N A194543 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n into parts p_i such that |p_i - p_j| >= k for i != j.
%C A194543 T(n,k) = 1 for n >= 0 and k >= n.
%C A194543 In general, column k > 0 is asymptotic to c^(1/4) * r * exp(2*sqrt(c*n)) / (2*sqrt(Pi*(1-r)*(r + k*(1-r))) * n^(3/4)), where r is the smallest real root of the equation r^k + r = 1 and c = k*log(r)^2/2 + polylog(2, 1-r). - _Vaclav Kotesovec_, Jan 02 2016
%H A194543 Alois P. Heinz, <a href="/A194543/b194543.txt">Rows n = 0..140</a>
%F A194543 G.f. of column k: Sum_{j>=0} x^(j*((j-1)*k/2+1))/Product_{i=1..j} (1-x^i).
%e A194543 T(7,3) = 3: [7], [6,1], [5,2].
%e A194543 T(23,6) = 11: [23], [22,1], [21,2], [20,3], [19,4], [18,5], [17,6], [16,7], [15,8], [15,7,1], [14,8,1].
%e A194543 Triangle begins:
%e A194543    1;
%e A194543    1, 1;
%e A194543    2, 1, 1;
%e A194543    3, 2, 1, 1;
%e A194543    5, 2, 2, 1, 1;
%e A194543    7, 3, 2, 2, 1, 1;
%e A194543   11, 4, 3, 2, 2, 1, 1;
%e A194543   15, 5, 3, 3, 2, 2, 1, 1;
%p A194543 b:= proc(n, i, k) option remember;
%p A194543       if n<0 then 0
%p A194543     elif n=0 then 1
%p A194543     else add(b(n-i-j, i+j, k), j=k..n-i)
%p A194543       fi
%p A194543     end:
%p A194543 T:= (n, k)-> `if`(n=0, 1, 0) +add(b(n-i, i, k), i=1..n):
%p A194543 seq(seq(T(n, k), k=0..n), n=0..20);
%t A194543 b[n_, i_, k_] := b[n, i, k] = If[n<0, 0, If[n == 0, 1, Sum[b[n-i-j, i+j, k], {j, k, n-i}]]]; T[n_, k_] := If[n == 0, 1, 0] + Sum[b[n-i, i, k], {i, 1, n}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Jan 19 2015, after _Alois P. Heinz_ *)
%Y A194543 Columns 0-8 give: A000041, A000009, A003114, A025157, A025158, A025159, A025160, A025161, A025162. T(n,0)-T(n,1) = A047967(n).
%K A194543 nonn,tabl
%O A194543 0,4
%A A194543 _Alois P. Heinz_, Aug 29 2011