cp's OEIS Frontend

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

%I A339282 #5 Nov 30 2020 21:40:11
%S A339282 1,2,2,4,14,10,11,84,168,98,30,522,2109,3004,1396,98,3426,24397,63094,
%T A339282 67660,25652,328,23404,274626,1142420,2119985,1805082,576010,1193,
%U A339282 165417,3065376,19230320,54916745,78809079,55503392,15282038,4459,1197934,34201068,311157620,1283360335,2761083930,3220245007,1932118328,467747416
%N A339282 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks with n colored elements using exactly k colors.
%C A339282 Unoriented version of A339228. Equivalence is up to reversal of all parts combined in series.
%e A339282 Triangle begins:
%e A339282     1;
%e A339282     2,     2;
%e A339282     4,    14,     10;
%e A339282    11,    84,    168,      98;
%e A339282    30,   522,   2109,    3004,    1396;
%e A339282    98,  3426,  24397,   63094,   67660,   25652;
%e A339282   328, 23404, 274626, 1142420, 2119985, 1805082, 576010;
%e A339282   ...
%o A339282 (PARI) \\ R(n, k) gives colorings using at most k colors as a vector.
%o A339282 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339282 B(n, Z)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); p}
%o A339282 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}
%o A339282 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])))}
%o A339282 {my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}
%Y A339282 Columns 1..2 are A339225, A339281.
%Y A339282 Row sums are A339283.
%Y A339282 Cf. A339228, A339280, A339285.
%K A339282 nonn,tabl
%O A339282 1,2
%A A339282 _Andrew Howroyd_, Nov 30 2020