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A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.

%I A339285 #7 Nov 30 2020 21:40:32
%S A339285 1,1,1,1,2,1,1,4,5,1,1,6,14,8,1,1,9,34,39,14,1,1,12,68,132,94,20,1,1,
%T A339285 16,126,370,447,202,30,1,1,20,212,887,1625,1275,398,40,1,1,25,340,
%U A339285 1911,4955,5985,3284,730,55,1,1,30,515,3765,13133,22608,19245,7649,1266,70,1
%N A339285 Triangle read by rows: T(n,k) is the number of unoriented series-parallel networks whose multigraph has n edges and k interior vertices, 0 <= k < n.
%C A339285 Unoriented version of A339231. Equivalence is up to reversal of all parts combined in series.
%F A339285 T(n,0) = T(n,n-1) = 1.
%F A339285 T(n,1) = A002620(n).
%F A339285 A339286(n) = Sum_{k=1..n-1} k*T(n,k).
%e A339285 Triangle begins:
%e A339285   1;
%e A339285   1,  1;
%e A339285   1,  2,   1;
%e A339285   1,  4,   5,    1;
%e A339285   1,  6,  14,    8,    1;
%e A339285   1,  9,  34,   39,   14,    1;
%e A339285   1, 12,  68,  132,   94,   20,    1;
%e A339285   1, 16, 126,  370,  447,  202,   30,   1;
%e A339285   1, 20, 212,  887, 1625, 1275,  398,  40,  1;
%e A339285   1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1;
%e A339285   ...
%e A339285 T(4,0) = 1: (o|o|o|o).
%e A339285 T(4,1) = 4: ((o|o)(o|o)), (o(o|o|o)), (o|o|oo), (o|o(o|o)).
%e A339285 T(4,2) = 5: (oo(o|o)), (o(o|o)o),  (o(o|oo)),  (oo|oo), (o|ooo).
%e A339285 T(4,3) = 1: (oooo).
%o A339285 (PARI)
%o A339285 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 A339285 SubPwr(p,e)={my(vars=variables(p)); substvec(p, vars, [v^e|v<-vars])}
%o A339285 BW(n, Z, W)={my(p=Z+O(x^2)); for(n=2, n, p=x*Ser(EulerMT(Vec(W*p^2/(1+W*p)+Z)))); p}
%o A339285 VertexWeighted(n, Z, W)={my(q=SubPwr(BW((n+1)\2, Z, W), 2), W2=SubPwr(W, 2), s=SubPwr(Z, 2)+W2*q^2/(1+W2*q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(W + W2*p); p=Z + x*Ser(EulerMT(Vec(t+(s-SubPwr(t, 2))/2))) - t); Vec(p+t-Z+BW(n, Z, W))/2}
%o A339285 T(n)={[Vecrev(p)|p<-VertexWeighted(n, x, y)]}
%o A339285 { my(A=T(12)); for(n=1, #A, print(A[n])) }
%Y A339285 Row sums are A339225.
%Y A339285 Cf. A002620, A339231, A339282, A339286.
%K A339285 nonn,tabl
%O A339285 1,5
%A A339285 _Andrew Howroyd_, Nov 30 2020