OFFSET
0,5
LINKS
Alois P. Heinz, Rows n = 1..140, flattened
Wikipedia, Partition (number theory)
FORMULA
Sum_{k=1..n} k * T(n,k) = A326656(n).
EXAMPLE
T(3,1) = 3: 3aaa, 2aa1a, 111aaa.
T(3,2) = 8: 3aab, 3abb, 2aa1b, 2ab1b, 2ab1a, 2bb1a, 111aab, 111abb.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 2;
0, 3, 8, 5;
0, 5, 22, 30, 13;
0, 7, 54, 129, 124, 42;
0, 11, 118, 428, 696, 525, 150;
0, 15, 248, 1293, 3108, 3830, 2358, 576;
0, 22, 490, 3483, 11595, 20720, 20535, 10661, 2266;
0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966;
...
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[n-t, Min[n-t, i-1], k] Binomial[k + t - 1, t]][i j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i] (-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved