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

%I A123543 #22 Jul 11 2025 03:03:48
%S A123543 0,1,2,14,201,4704,160890,7538040,462869190,36055948320,3474195588360,
%T A123543 405786523413600,56502317464777800,9248640671612865600,
%U A123543 1758505909558569771600,384399253128691423022400,95737858067835530264718000,26952922550751326069548608000
%N A123543 Number of connected labeled 2-regular pseudodigraphs (multiple arcs and loops allowed) of order n.
%D A123543 R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
%H A123543 Alois P. Heinz, <a href="/A123543/b123543.txt">Table of n, a(n) for n = 0..253</a> (first 49 terms from R. W. Robinson)
%F A123543 E.g.f.: log(1 + Sum_{k>0} A000681(k)*x^k/k!). - _Andrew Howroyd_, Sep 09 2018
%F A123543 a(n) ~ 2 * sqrt(Pi) * n^(2*n + 1/2) / exp(2*n - 1/2). - _Vaclav Kotesovec_, Jul 11 2025
%p A123543 b:= proc(n) option remember; `if`(n<2, 1,
%p A123543       n^2*b(n-1)-n*(n-1)^2*b(n-2)/2)
%p A123543     end:
%p A123543 a:= proc(n) option remember; `if`(n=0, 0, b(n)-
%p A123543       add(j*binomial(n, j)*b(n-j)*a(j), j=1..n-1)/n)
%p A123543     end:
%p A123543 seq(a(n), n=0..17);  # _Alois P. Heinz_, Mar 22 2025
%t A123543 m = 16;
%t A123543 a681[n_] := n!*HypergeometricPFQ[{1/2, -n}, {}, -2]/2^n;
%t A123543 egf = Log[1 + Sum[a681[k] x^k/k!, {k, 1, m}]];
%t A123543 CoefficientList[egf + O[x]^m, x] Range[0, m-1]! (* _Jean-François Alcover_, Aug 26 2019 *)
%o A123543 (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
%Y A123543 Connected version of A000681.
%Y A123543 First column of A307804.
%Y A123543 Cf. A123544.
%K A123543 nonn
%O A123543 0,3
%A A123543 _N. J. A. Sloane_, Nov 13 2006