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A218698 Number T(n,k) of ways to divide the partitions of n into nonempty consecutive subsequences each of which contains only equal parts and parts from distinct subsequences differ by at least k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A218698 #25 Dec 10 2020 17:42:11
%S A218698 1,1,1,3,2,2,6,3,2,2,14,5,4,3,3,27,7,4,3,2,2,60,11,8,6,5,4,4,117,15,8,
%T A218698 6,4,3,2,2,246,22,13,9,8,6,5,4,4,490,30,15,12,8,7,5,4,3,3,1002,42,22,
%U A218698 14,12,9,8,6,5,4,4,1998,56,24,16,12,10,7,6,4,3,2,2
%N A218698 Number T(n,k) of ways to divide the partitions of n into nonempty consecutive subsequences each of which contains only equal parts and parts from distinct subsequences differ by at least k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A218698 T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A000005(n) for k >= n.  For k>0: T(n,k) = number of partitions of n in which any two distinct parts differ by at least k, or, equivalently, T(n,k) = number of partitions of n in which each part, with the possible exception of the largest, occurs at least k times.
%H A218698 Alois P. Heinz, <a href="/A218698/b218698.txt">Rows n = 0..140, flattened</a>
%F A218698 G.f. of column k: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(k*i)/(1-x^i)).
%e A218698 T(4,0) = 14: [[1],[1],[1],[1]], [[1,1],[1],[1]], [[1],[1,1],[1]], [[1,1,1],[1]], [[1],[1],[1,1]], [[1,1],[1,1]], [[1],[1,1,1]], [[1,1,1,1]], [[1],[1],[2]], [[1,1],[2]], [[2],[2]], [[2,2]], [[1],[3]], [[4]].
%e A218698 T(4,1) = 5: [[1,1,1,1]], [[1,1],[2]], [[2,2]], [[1],[3]], [[4]].
%e A218698 T(4,2) = 4: [[1,1,1,1]], [[2,2]], [[1],[3]], [[4]].
%e A218698 T(4,3) = T(4,4) = A000005(4) = 3: [[1,1,1,1]], [[2,2]], [[4]].
%e A218698 Triangle T(n,k) begins:
%e A218698     1;
%e A218698     1,  1;
%e A218698     3,  2,  2;
%e A218698     6,  3,  2,  2;
%e A218698    14,  5,  4,  3,  3;
%e A218698    27,  7,  4,  3,  2,  2;
%e A218698    60, 11,  8,  6,  5,  4,  4;
%e A218698   117, 15,  8,  6,  4,  3,  2,  2;
%e A218698   ...
%p A218698 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A218698        b(n, i-1, k) +add(b(n-i*j, i-k, k), j=1..n/i)))
%p A218698     end:
%p A218698 T:= (n, k)-> b(n, n, k):
%p A218698 seq(seq(T(n,k), k=0..n), n=0..12);
%t A218698 b[n_, i_, k_] :=  b[n, i, k] =  If[n == 0, 1, If[i < 1, 0,  b[n, i - 1, k] + Sum[b[n - i*j, i - k, k], {j, 1, n/i}]]]; T[n_, k_] := b[n, n, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 27 2013, translated from Maple *)
%Y A218698 Columns k=0-10 give: A006951, A000041, A116931, A116932, A218699, A218700, A218701, A218702, A218703, A218704, A218705.
%Y A218698 Main diagonal gives: A000005.
%Y A218698 T(2n,n) gives A319776.
%K A218698 nonn,tabl
%O A218698 0,4
%A A218698 _Alois P. Heinz_, Nov 04 2012