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.
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, 16, 126, 370, 447, 202, 30, 1, 1, 20, 212, 887, 1625, 1275, 398, 40, 1, 1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1, 1, 30, 515, 3765, 13133, 22608, 19245, 7649, 1266, 70, 1
Offset: 1
Examples
Triangle begins: 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, 16, 126, 370, 447, 202, 30, 1; 1, 20, 212, 887, 1625, 1275, 398, 40, 1; 1, 25, 340, 1911, 4955, 5985, 3284, 730, 55, 1; ... T(4,0) = 1: (o|o|o|o). T(4,1) = 4: ((o|o)(o|o)), (o(o|o|o)), (o|o|oo), (o|o(o|o)). T(4,2) = 5: (oo(o|o)), (o(o|o)o), (o(o|oo)), (oo|oo), (o|ooo). T(4,3) = 1: (oooo).
Programs
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PARI
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)} SubPwr(p,e)={my(vars=variables(p)); substvec(p, vars, [v^e|v<-vars])} 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} 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} T(n)={[Vecrev(p)|p<-VertexWeighted(n, x, y)]} { my(A=T(12)); for(n=1, #A, print(A[n])) }
Comments