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A327639
Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.
8
1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 15, 16, 6, 1, 10, 45, 88, 76, 24, 1, 14, 93, 282, 420, 302, 84, 1, 21, 223, 1052, 2489, 3112, 1970, 498, 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220, 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108
OFFSET
0,6
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
Row n is the inverse binomial transform of the n-th row of array A323718.
LINKS
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A323718(n,i).
T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.
Sum_{k=1..n-1} k * T(n,k) = A327646(n).
Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.
EXAMPLE
T(4,0) = 1: 4
T(4,1) = 4: T(4,2) = 6: T(4,3) = 3:
4-> 31 4-> 31 -> 211 4-> 31 -> 211 -> 1111
4-> 22 4-> 31 -> 1111 4-> 22 -> 112 -> 1111
4-> 211 4-> 22 -> 112 4-> 22 -> 211 -> 1111
4-> 1111 4-> 22 -> 211
4-> 22 -> 1111
4-> 211-> 1111
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 1;
1, 4, 6, 3;
1, 6, 15, 16, 6;
1, 10, 45, 88, 76, 24;
1, 14, 93, 282, 420, 302, 84;
1, 21, 223, 1052, 2489, 3112, 1970, 498;
1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220;
1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-1)), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[0, n - 1] }] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give A000012, A000065, A327769.
Row sums give A327644.
T(2n,n) gives A327645.
Sequence in context: A351790 A322266 A190284 * A273891 A034870 A324224
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved