OFFSET
1,4
COMMENTS
See A330655 for the definition of a balanced reduced multisystem.
A balanced reduced multisystem of weight n with atoms of k colors corresponds with a rooted tree with n leaves of k colors with all leaves at the same depth and at least one node at each level of the tree having more than one child. The final condition is needed to ensure that the number of such trees is finite.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
EXAMPLE
Triangle begins:
1;
1, 1;
2, 6, 4;
6, 37, 63, 32;
20, 262, 870, 1064, 436;
90, 2217, 12633, 27824, 26330, 9012;
468, 21882, 201654, 710712, 1163320, 895608, 262760;
...
The T(3,2) = 6 balanced reduced multisystems are: {1,1,2}, {1,2,2}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}.
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1, n-1)); v=EulerT(v)); u}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 30 2019
STATUS
approved