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
Alois P. Heinz, Rows n = 0..170, flattened
Wikipedia, Iverson bracket
Wikipedia, Partition (number theory)
FORMULA
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
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 20 2019
STATUS
approved