OFFSET
1,3
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 A327618.
LINKS
Alois P. Heinz, Rows n = 1..200, flattened
Wikipedia, Partition (number theory)
FORMULA
EXAMPLE
T(4,0) = 1:
4 (1 part).
T(4,1) = 11 = 2 + 2 + 3 + 4:
4-> 31 (2 parts)
4-> 22 (2 parts)
4-> 211 (3 parts)
4-> 1111 (4 parts)
T(4,2) = 21 = 3 + 4 + 3 + 3 + 4 + 4:
4-> 31 -> 211 (3 parts)
4-> 31 -> 1111 (4 parts)
4-> 22 -> 112 (3 parts)
4-> 22 -> 211 (3 parts)
4-> 22 -> 1111 (4 parts)
4-> 211-> 1111 (4 parts)
T(4,3) = 12 = 4 + 4 + 4:
4-> 31 -> 211 -> 1111 (4 parts)
4-> 22 -> 112 -> 1111 (4 parts)
4-> 22 -> 211 -> 1111 (4 parts)
Triangle T(n,k) begins:
1;
1, 2;
1, 5, 3;
1, 11, 21, 12;
1, 19, 61, 74, 30;
1, 34, 205, 461, 432, 144;
1, 53, 474, 1652, 2671, 2030, 588;
1, 85, 1246, 6795, 17487, 23133, 15262, 3984;
1, 127, 2723, 20966, 76264, 148134, 158452, 88194, 19980;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
`if`(k=0, [1, 1], `if`(i<2, 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), 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..n-1), n=1..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i < 2, 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], 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[T[n, k], {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Jan 07 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 19 2019
STATUS
approved