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

0,3

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)

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

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 *)

Prog

(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

Crossrefs

Connected version of A000681.

First column of A307804.

Cf. A123544.

Sequence in context: A213977 A322196 A102224 * A279452 A262008 A054652

Adjacent sequences: A123540 A123541 A123542 * A123544 A123545 A123546

Keyword

nonn

Author

N. J. A. Sloane, Nov 13 2006

Status

approved