OFFSET
1,4
COMMENTS
In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
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 A327622.
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
Wikipedia, Partition (number theory)
FORMULA
T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327622(n,i).
T(n+1,n-1) = 1 for n >= 1.
EXAMPLE
T(4,0) = 1:
4 (1 part).
T(4,1) = 2:
4-> 31 (2 parts)
T(4,2) = 3:
4-> 31 -> 211 (3 parts)
Triangle T(n,k) begins:
1;
1;
1, 2;
1, 2, 3;
1, 4, 6, 4;
1, 7, 13, 12, 5;
1, 9, 30, 52, 35, 6;
1, 12, 61, 137, 156, 72, 7;
1, 17, 121, 384, 638, 548, 196, 8;
1, 24, 210, 880, 1983, 2442, 1543, 400, 9;
1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
(h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
end:
T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..max(0, n-2)), n=1..14);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {0, 0}, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][ b[i, i, k - 1]]]]]];
T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 0, Max[0, n - 2]}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 19 2019
STATUS
approved