A339299 -id:A339299 - cp's OEIS frontend

A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

1, 2, 12, 108, 1380, 22440, 446040, 10461360, 282970800, 8670594240, 296850597120, 11230473925440, 465262142304960, 20948652798353280, 1018583225567107200, 53190962586022060800, 2969038807022050963200, 176410305542414738995200, 11116489894884127122969600

Offset: 1

Andrew Howroyd, Dec 22 2020

nonn

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements labeled 1..n.

			a(3) = 12 because there are 2 unlabeled structures each of which can be labeled in 6 ways. The unlabeled structures are (ooo) and (o|oo).

A048172 is the case with multiple unit elements in parallel allowed.
A058381 is the case that order is not significant in series configurations.
Main diagonal of A339297.
Cf. A339290 (unlabeled), A339299, A339300.

\\ Note giving Z=exp(x)-1 gives A048172.
seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(p))}

seq(n)={my(A=O(x*x^n)); Vec(serlaplace(subst(serreverse(log(1+x+A) - x^2/(1+x)), x, log(1+x+A))))}

a(n) = A339299(n) + A339300(n).
E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.
E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.

A339300 Number of essentially parallel oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 6, 36, 540, 8400, 169680, 3966480, 107518320, 3295283040, 112888369440, 4272403544640, 177061349424960, 7974538914101760, 387840385867334400, 20257533315635616000, 1130954856127948051200, 67208532822729871372800, 4235759061057115720128000

Offset: 1

Andrew Howroyd, Dec 22 2020

nonn

See A339301 for additional details.

A048174 is the case with multiple edges in parallel allowed.
A058379 is the case that order is not significant in series configurations.
Cf. A339289 (unlabeled), A339299, A339301.

seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(1-1/(1+p)))}

E.g.f. (1 + x)*exp(S(x)) - S(x) - 1 where S(x) is the e.g.f. of A339299.
E.g.f.: B(x)/(1 + B(x)) where B(x) is the e.g.f. of A339301.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048174.

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A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

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A339300 Number of essentially parallel oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

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Author

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PARI

Formula