A339282 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks with n colored elements using exactly k colors.

1, 2, 2, 4, 14, 10, 11, 84, 168, 98, 30, 522, 2109, 3004, 1396, 98, 3426, 24397, 63094, 67660, 25652, 328, 23404, 274626, 1142420, 2119985, 1805082, 576010, 1193, 165417, 3065376, 19230320, 54916745, 78809079, 55503392, 15282038, 4459, 1197934, 34201068, 311157620, 1283360335, 2761083930, 3220245007, 1932118328, 467747416

Offset: 1

Author

Andrew Howroyd, Nov 30 2020

Keywords

• nonn
• tabl

Comments

Unoriented version of A339228. Equivalence is up to reversal of all parts combined in series.

Examples

			Triangle begins:
    1;
    2,     2;
    4,    14,     10;
   11,    84,    168,      98;
   30,   522,   2109,    3004,    1396;
   98,  3426,  24397,   63094,   67660,   25652;
  328, 23404, 274626, 1142420, 2119985, 1805082, 576010;
  ...
		

Crossrefs

Columns 1..2 are A339225, A339281.

Row sums are A339283.

Cf. A339228, A339280, A339285.

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)}
B(n, Z)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); p}
R(n,k)={my(Z=k*x, q=subst(B((n+1)\2, Z), x, x^2), s=subst(Z,x,x^2)+q^2/(1+q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(1 + p); p=Z + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+t-Z+B(n,Z))/2}
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]))}