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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.
12
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, 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, 14, 12, 9, 8, 6, 5, 4, 4, 1998, 56, 24, 16, 12, 10, 7, 6, 4, 3, 2, 2
OFFSET
0,4
COMMENTS
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.
LINKS
FORMULA
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)).
EXAMPLE
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]].
T(4,1) = 5: [[1,1,1,1]], [[1,1],[2]], [[2,2]], [[1],[3]], [[4]].
T(4,2) = 4: [[1,1,1,1]], [[2,2]], [[1],[3]], [[4]].
T(4,3) = T(4,4) = A000005(4) = 3: [[1,1,1,1]], [[2,2]], [[4]].
Triangle T(n,k) begins:
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, 6, 4, 3, 2, 2;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1, k) +add(b(n-i*j, i-k, k), j=1..n/i)))
end:
T:= (n, k)-> b(n, n, k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
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 *)
CROSSREFS
Main diagonal gives: A000005.
T(2n,n) gives A319776.
Sequence in context: A093055 A285733 A106335 * A352836 A065474 A272332
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 04 2012
STATUS
approved