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%I #8 Nov 30 2020 08:56:48
%S 1,2,3,5,22,19,15,146,321,195,48,970,4116,5972,2791,167,6601,48245,
%T 125778,135235,51303,602,46012,546570,2281528,4238415,3609966,1152019,
%U 2256,328188,6118320,38437972,109815445,157612413,111006329,30564075
%N Triangle read by rows: T(n,k) is the number of oriented series-parallel networks with n colored elements using exactly k colors.
%C A series configuration is a unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is a unit element or a multiset of two or more series configurations. 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 Triangle begins:
%e 1;
%e 2, 3;
%e 5, 22, 19;
%e 15, 146, 321, 195;
%e 48, 970, 4116, 5972, 2791;
%e 167, 6601, 48245, 125778, 135235, 51303;
%e 602, 46012, 546570, 2281528, 4238415, 3609966, 1152019;
%e ...
%o (PARI) \\ R(n,k) gives colorings using at most k colors as a vector.
%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o R(n,k)={my(Z=k*x, p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+Z)))); Vec(p)}
%o 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 {my(T=M(8)); for(n=1, #T~, print(T[n, 1..n]))}
%Y Columns 1..2 are A003430, A339227.
%Y Row sums are A339229.
%Y Main diagonal is A048172.
%Y Cf. A339226, A339228.
%K nonn,tabl
%O 1,2
%A _Andrew Howroyd_, Nov 28 2020
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