OFFSET
0,6
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition (number theory)
FORMULA
Sum_{k=1..n} k * T(n,k) = A327557(n).
EXAMPLE
T(3,2) = 6; 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 2, 6, 5;
0, 2, 15, 27, 15;
0, 3, 32, 102, 124, 52;
0, 4, 65, 319, 656, 600, 203;
0, 5, 124, 897, 2780, 4210, 3084, 877;
0, 6, 230, 2346, 10305, 23040, 27567, 16849, 4140;
0, 8, 414, 5818, 34864, 108135, 188284, 186095, 97640, 21147;
...
MAPLE
C:= binomial:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
c = Binomial;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *)
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 13 2019
STATUS
approved