OFFSET
0,5
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
EXAMPLE
Triangle T(n,k) begins:
1;
0, 1;
0, 3;
0, 7, 3;
0, 31, 16;
0, 121, 125;
0, 831, 711, 60;
0, 5041, 5915, 525;
0, 42911, 46264, 6328;
0, 364561, 438681, 67788;
0, 3742453, 4371085, 753420, 12600;
...
MAPLE
with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
select(x-> nops({x[]})=k, partition(n))):
seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..14);
# second Maple program:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(x^signum(j)*b(n-i*j, i-1)*
combinat[multinomial](n, n-i*j, i$j), j=0..n/i))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..14);
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i<1, 0, Sum[x^Sign[j]*b[n - i*j, i-1]*multinomial[n, Join[{n-i*j}, Table[i, {j}]]], {j, 0, n/i}]]]];
T[n_] := CoefficientList[b[n, n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, May 06 2020, after 2nd Maple program *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 25 2019
STATUS
approved