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A339228
Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.
10
1, 2, 3, 5, 22, 19, 15, 146, 321, 195, 48, 970, 4116, 5972, 2791, 167, 6601, 48245, 125778, 135235, 51303, 602, 46012, 546570, 2281528, 4238415, 3609966, 1152019, 2256, 328188, 6118320, 38437972, 109815445, 157612413, 111006329, 30564075
OFFSET
1,2
COMMENTS
A series configuration is a unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is a unit element or a multiset of two or more series configurations. 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;
2, 3;
5, 22, 19;
15, 146, 321, 195;
48, 970, 4116, 5972, 2791;
167, 6601, 48245, 125778, 135235, 51303;
602, 46012, 546570, 2281528, 4238415, 3609966, 1152019;
...
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=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); 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
Columns 1..2 are A003430, A339227.
Row sums are A339229.
Main diagonal is A048172.
Sequence in context: A093522 A352124 A093499 * A013639 A047995 A244862
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Nov 28 2020
STATUS
approved