A339152
Number of essentially parallel nonequivalent series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
1, 0, 0, 1, 1, 3, 4, 9, 16, 33, 63, 131, 261, 545, 1123, 2359, 4948, 10502, 22307, 47731, 102367, 220600, 476626, 1033450, 2246252, 4895935, 10694744, 23414838, 51364180, 112891831, 248548836, 548123924, 1210612692, 2677682900, 5930586249, 13151963729, 29201456634
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)), (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(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); P}
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}
A339154
Number of essentially series oriented series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
0, 1, 1, 1, 3, 6, 14, 30, 70, 165, 397, 961, 2368, 5875, 14722, 37134, 94312, 240823, 618147, 1593606, 4125218, 10717064, 27934867, 73032798, 191464677, 503218042, 1325678981, 3499913710, 9258627528, 24538328431, 65147600774, 173243773337, 461400769439
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) = 3: (ooooo), (o(oo|oo)), ((oo|oo)o).
a(6) = 6: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o).
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=O(x^2)); for(n=2, n, p=x+x*Ser(EulerT(Vec(p, 1-n))); p=p^2/(1+p)); Vec(p, -n)}
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