OFFSET
0,6
COMMENTS
T(n,k) is the sum of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts that form a multiset of size k.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Iverson bracket
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
EXAMPLE
T(4,2) = 10: aaab, aaba, aabb, abaa, abab, abba, baaa, baab, baba, bbaa.
T(4,3) = 12: aabc, aacb, abac, abca, acab, acba, baac, baca, bcaa, caab, caba, cbaa.
T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 2;
0, 1, 3, 6;
0, 1, 10, 12, 24;
0, 1, 15, 50, 60, 120;
0, 1, 41, 180, 300, 360, 720;
0, 1, 63, 497, 1260, 2100, 2520, 5040;
0, 1, 162, 1484, 6496, 10080, 16800, 20160, 40320;
...
MAPLE
b:= proc(n, i, t) option remember;
`if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
end:
T:= (n, k)-> `if`(n*k=0, `if`(n=k, 1, 0), n!*b(n, 1, k)):
seq(seq(T(n, k), k=0..n), n=0..12);
# second Maple program:
b:= proc(n, i) option remember; expand(
`if`(n=0, 1, `if`(i<1, 0, add(x^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..n))(b(n$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]]; t[n_, k_] := If[n*k == 0, If[n == k, 1, 0], n!*b[n, 1, k]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from first Maple *)
PROG
(PARI)
T(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
{my(t=T(10)); for(n=1, #t, for(k=0, n-1, print1(polcoeff(t[n], k), ", ")); print)} \\ Andrew Howroyd, Dec 20 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 21 2013
STATUS
approved