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A339231 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

%I A339231 #10 Jan 18 2022 19:37:05
%S A339231 1,1,1,1,3,1,1,6,7,1,1,10,23,13,1,1,15,59,69,22,1,1,21,124,249,172,34,
%T A339231 1,1,28,234,711,853,378,50,1,1,36,402,1733,3175,2487,755,70,1,1,45,
%U A339231 650,3755,9767,11813,6431,1400,95,1,1,55,995,7443,26043,44926,38160,15098,2445,125,1
%N A339231 Triangle read by rows: T(n,k) is the number of oriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.
%C A339231 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 whose representation as a multigraph has k interior vertices, with elements corresponding to edges. Parallel configurations do not increase the interior vertex count and series configurations increase it by one less than the number of parts.
%F A339231 T(n,0) = T(n,n-1) = 1.
%F A339231 T(n,1) = binomial(n,2).
%F A339231 T(n+2,n) = A002623(n).
%F A339231 Sum_{k=1..n-1} k*T(n,k) = A339232(n).
%e A339231 Triangle begins:
%e A339231   1;
%e A339231   1,  1;
%e A339231   1,  3,   1;
%e A339231   1,  6,   7,   1;
%e A339231   1, 10,  23,  13,   1;
%e A339231   1, 15,  59,  69,  22,   1;
%e A339231   1, 21, 124, 249, 172,  34,  1;
%e A339231   1, 28, 234, 711, 853, 378, 50, 1;
%e A339231   ...
%e A339231 In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
%e A339231 T(4,0) = 1: (o|o|o|o).
%e A339231 T(4,1) = 6: ((o|o)(o|o)), (o(o|o|o)), ((o|o|o)o), (o|o|oo), (o|o(o|o)), (o|(o|o)o).
%e A339231 T(4,2) = 7: (oo(o|o)), (o(o|o)o), ((o|o)oo),  (o(o|oo)), ((o|oo)o),  (oo|oo), (o|ooo).
%e A339231 T(4,3) = 1: (oooo).
%e A339231 The graph of (oo(o|o)) has 4 edges (elements) and 2 interior vertices as shown below:
%e A339231       A---o---o===Z (where === is a double edge).
%o A339231 (PARI)
%o A339231 EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, [v^i|v<-vars])/i ))-1)}
%o A339231 VertexWeighted(n, W)={my(Z=x, p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1 + W*p) + Z)))); Vec(p)}
%o A339231 T(n)={[Vecrev(p)|p<-VertexWeighted(n,y)]}
%o A339231 { my(A=T(12)); for(n=1, #A, print(A[n])) }
%Y A339231 Row sums are A003430.
%Y A339231 Cf. A002623, A339228, A339230, A339232.
%K A339231 nonn,tabl
%O A339231 1,5
%A A339231 _Andrew Howroyd_, Nov 29 2020