cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

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.

Examples

			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).
		

Crossrefs

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.

Programs

  • PARI
    \\ 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))}
    
  • PARI
    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))))}

Formula

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.