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

1,3

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.

Links

Table of n, a(n) for n=1..45.

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

Column 1 is A339290.

Main diagonal is A339301.

Row sums are A339298.

Cf. A339228.

Sequence in context: A228154 A275279 A109767 * A196061 A342582 A131121

Adjacent sequences: A339294 A339295 A339296 * A339298 A339299 A339300

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 22 2020

Status

approved