A339295 Number of essentially parallel unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

1, 0, 1, 2, 4, 10, 25, 69, 197, 589, 1806, 5685, 18168, 58905, 192904, 637294, 2119994, 7094961, 23865782, 80642017, 273571625, 931389949, 3181184007, 10897272983, 37429033777, 128874546753, 444744161951, 1538030244174, 5329246656885, 18499283612755

Offset

1,4

Comments

See A339296 for additional details.

Links

Table of n, a(n) for n=1..30.

Formula

a(n) = (A339289(n) + A339292(n)) / 2.

Example

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(3) = 1: (o|oo).
a(4) = 2: (o|ooo), (oo|oo).
a(5) = 4: (o|oooo), (o|o(o|oo)), (oo|ooo), (o|oo|oo).

Prog

(PARI) \\ here B(n) gives A339290 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); p}
seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=Z+O(x^2), t=0); forstep(n=2, n, 2, t=q*(1 + p); p=Z + (1 + Z)*x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2, -n-1))) - t); Vec(p+1-1/(1+B(n, Z)))/2}

Crossrefs

Cf. A339224, A339289 (oriented), A339292 (achiral), A339294, A339296.

Sequence in context: A027432 A032128 A052829 * A001998 A005817 A302093

Adjacent sequences: A339292 A339293 A339294 * A339296 A339297 A339298

Keyword

nonn

Author

Andrew Howroyd, Dec 07 2020

Status

approved