A339228 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.

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

Author

Andrew Howroyd, Nov 28 2020

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

Crossrefs

Columns 1..2 are A003430, A339227.

Row sums are A339229.

Main diagonal is A048172.

Cf. A339226, A339228.

Programs

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]))}