A339298 -id:A339298 - cp's OEIS frontend

A339297 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements and without multiple unit elements in parallel using exactly k colors.

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, 306, 68427, 1524969, 10652086, 32760180, 49718640, 36614760, 10461360, 930, 414294, 14382720, 146464740, 652517010, 1487453760, 1816345440, 1131883200, 282970800

Offset: 1

Andrew Howroyd, Dec 22 2020

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.

			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;
  ...

Column 1 is A339290.
Main diagonal is A339301.
Row sums are A339298.
Cf. A339228.

\\ 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]))}

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A339297 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements and without multiple unit elements in parallel using exactly k colors.

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