A339151
Number of essentially series nonequivalent series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
0, 1, 1, 1, 2, 3, 6, 11, 21, 41, 81, 164, 333, 684, 1417, 2965, 6229, 13182, 28019, 59862, 128376, 276408, 597041, 1293763, 2811181, 6124479, 13374201, 29270703, 64191331, 141041901, 310444472, 684445543, 1511345542, 3342101662, 7400605768, 16408670095
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 2: (ooooo), (o(oo|oo)).
a(6) = 3: (oooooo), (oo(oo|oo)), (o(oo|ooo)).
a(7) = 6: (ooooooo), (ooo(oo|oo)), (oo(oo|ooo)), (o(oo|oooo)), (o(ooo|ooo)), (o(oo|oo|oo)).
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(S=vector(n), P=vector(n)); P[1]=1; for(n=2, #S, my(t=EulerT(S[1..n])[n]); S[n]=EulerT(P[1..n])[n]; P[n]=t); S}
A339153
Number of nonequivalent series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 10, 20, 37, 74, 144, 295, 594, 1229, 2540, 5324, 11177, 23684, 50326, 107593, 230743, 497008, 1073667, 2327213, 5057433, 11020414, 24068945, 52685541, 115555511, 253933732, 558993308, 1232569467, 2721958234, 6019784562, 13331192017, 29560633824
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 2: (oooo), (oo|oo).
a(5) = 3: (ooooo), (o(oo|oo)), (oo|ooo).
a(6) = 6: (oooooo), (oo(oo|oo)), (o(oo|ooo)), (oo|oooo), (ooo|ooo), (oo|oo|oo).
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(S=vector(n), P=vector(n)); P[1]=1; for(n=2, #S, my(t=EulerT(S[1..n])[n]); S[n]=EulerT(P[1..n])[n]; P[n]=t); S+P}
A339155
Number of essentially parallel oriented series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
1, 0, 0, 1, 1, 3, 5, 13, 29, 70, 165, 409, 1001, 2505, 6278, 15904, 40447, 103567, 266229, 687668, 1782573, 4637731, 12103112, 31679212, 83135973, 218713492, 576683119, 1523740365, 4033915677, 10698680606, 28422818782, 75629586540, 201539697208, 537818080714
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(4) = 1: (oo|oo).
a(5) = 1: (oo|ooo).
a(6) = 3: (oo|oooo), (ooo|ooo), (oo|oo|oo).
a(7) = 4: (oo|ooooo), (oo|o(oo|oo)), (oo|(oo|oo)o), (ooo|oooo), (oo|oo|ooo).
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=x+O(x^2)); for(n=2, n, p=x+x*Ser(EulerT(Vec(p^2/(1+p), -n)))); Vec(1 - 1/(1+p))}
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