A339156
Number of oriented series-parallel networks with n elements and without unit elements in parallel.
Original entry on oeis.org
1, 1, 1, 2, 4, 9, 19, 43, 99, 235, 562, 1370, 3369, 8380, 21000, 53038, 134759, 344390, 884376, 2281274, 5907791, 15354795, 40037979, 104712010, 274600650, 721931534, 1902362100, 5023654075, 13292543205, 35237009037, 93570419556, 248873359877, 662940466647
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) = 4: (ooooo), (o(oo|oo)), ((oo|oo)o), (oo|ooo).
a(6) = 9: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o), (oo|oooo), (ooo|ooo), (oo|oo).
-
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(p)}
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)).
-
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}
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).
-
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}
Showing 1-3 of 3 results.
Comments