OFFSET
0,6
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
Sum_{k=1..n} k * T(n,k) = A327595(n).
EXAMPLE
T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
T(3,3) = 8: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 3, 10, 8;
0, 3, 27, 54, 31;
0, 5, 70, 255, 336, 147;
0, 11, 223, 1222, 2692, 2580, 899;
0, 13, 508, 4467, 15512, 25330, 19566, 5777;
0, 19, 1193, 15540, 78819, 194075, 248976, 160377, 41024;
...
MAPLE
C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
b(n-i*j, min(n-i*j, i-1), k, p+j)/j!*C(C(k+i-1, i), j), j=0..n/i)))
end:
T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p+j]/j!*c[c[k + i - 1, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 14 2019
STATUS
approved