A123543 Number of connected labeled 2-regular pseudodigraphs (multiple arcs and loops allowed) of order n.

0, 1, 2, 14, 201, 4704, 160890, 7538040, 462869190, 36055948320, 3474195588360, 405786523413600, 56502317464777800, 9248640671612865600, 1758505909558569771600, 384399253128691423022400, 95737858067835530264718000, 26952922550751326069548608000

Offset: 0

Author

N. J. A. Sloane, Nov 13 2006

Keywords

• nonn

References

• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..253 (first 49 terms from R. W. Robinson)

Crossrefs

Connected version of A000681.

First column of A307804.

Cf. A123544.

Programs

Maple

b:= proc(n) option remember; `if`(n<2, 1,
      n^2*b(n-1)-n*(n-1)^2*b(n-2)/2)
    end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-
      add(j*binomial(n, j)*b(n-j)*a(j), j=1..n-1)/n)
    end:
seq(a(n), n=0..17);  # Alois P. Heinz, Mar 22 2025

Mathematica

m = 16;
a681[n_] := n!*HypergeometricPFQ[{1/2, -n}, {}, -2]/2^n;
egf = Log[1 + Sum[a681[k] x^k/k!, {k, 1, m}]];
CoefficientList[egf + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Aug 26 2019 *)

PARI

seq(n)={Vec(serlaplace(log(serlaplace(exp(x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n))))), -(n+1))}; \\ Andrew Howroyd, Sep 09 2018

Formula

E.g.f.: log(1 + Sum_{k>0} A000681(k)*x^k/k!). - Andrew Howroyd, Sep 09 2018

a(n) ~ 2 * sqrt(Pi) * n^(2*n + 1/2) / exp(2*n - 1/2). - Vaclav Kotesovec, Jul 11 2025