OFFSET
0,4
COMMENTS
Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
EXAMPLE
Triangle T(n,k) begins:
1;
1, 1;
3, 2, 1;
10, 9, 3, 1;
47, 40, 18, 4, 1;
246, 235, 100, 30, 5, 1;
1602, 1476, 705, 200, 45, 6, 1;
11481, 11214, 5166, 1645, 350, 63, 7, 1;
95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
...
MAPLE
with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
`if`(i=k, 0, b(n-i, min(n-i, i), k)/i!))
end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 25 2019
STATUS
approved