OFFSET
1,3
COMMENTS
A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. T(n, k) is the number of series or parallel configurations with n unit elements of k colors using each color at least once.
EXAMPLE
Triangle begins:
1;
1, 2;
2, 12, 12;
5, 64, 162, 108;
13, 354, 1734, 2760, 1380;
36, 1992, 16977, 48716, 56100, 22440;
103, 11538, 161691, 746316, 1488240, 1338120, 446040;
...
PROG
(PARI) \\ R(n, k) gives colorings using at most k colors as a vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(Z=k*x, p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}
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(8)); for(n=1, #T~, print(T[n, 1..n]))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 22 2020
STATUS
approved