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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.

%I A339297 #6 Dec 22 2020 18:26:01
%S A339297 1,1,2,2,12,12,5,64,162,108,13,354,1734,2760,1380,36,1992,16977,48716,
%T A339297 56100,22440,103,11538,161691,746316,1488240,1338120,446040,306,68427,
%U A339297 1524969,10652086,32760180,49718640,36614760,10461360,930,414294,14382720,146464740,652517010,1487453760,1816345440,1131883200,282970800
%N 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.
%C A339297 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.
%e A339297 Triangle begins:
%e A339297     1;
%e A339297     1,     2;
%e A339297     2,    12,     12;
%e A339297     5,    64,    162,    108;
%e A339297    13,   354,   1734,   2760,    1380;
%e A339297    36,  1992,  16977,  48716,   56100,   22440;
%e A339297   103, 11538, 161691, 746316, 1488240, 1338120, 446040;
%e A339297   ...
%o A339297 (PARI) \\ R(n, k) gives colorings using at most k colors as a vector.
%o A339297 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339297 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)}
%o A339297 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 A339297 {my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}
%Y A339297 Column 1 is A339290.
%Y A339297 Main diagonal is A339301.
%Y A339297 Row sums are A339298.
%Y A339297 Cf. A339228.
%K A339297 nonn,tabl
%O A339297 1,3
%A A339297 _Andrew Howroyd_, Dec 22 2020