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A053549 Number of labeled rooted connected graphs.

%I A053549 #16 Sep 08 2022 08:45:00
%S A053549 0,1,2,12,152,3640,160224,13063792,2012388736,596666619648,
%T A053549 344964885948160,392058233038486784,880255154481199466496,
%U A053549 3916538634445633156373504,34603083354426212294072477696,607915214065957203519146330173440
%N A053549 Number of labeled rooted connected graphs.
%D A053549 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 10, R_p.
%D A053549 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20, G(x).
%H A053549 T. D. Noe, <a href="/A053549/b053549.txt">Table of n, a(n) for n=0..50</a>
%F A053549 E.g.f.: (Sum_{n>1} 2^binomial(n, 2)*x^n/(n-1)!)/(Sum_{n>=0} 2^binomial(n, 2)*x^n/n!).
%F A053549 a(n) = n * A001187(n).
%p A053549 add(2^binomial(n,2)*x^n/(n-1)!,n=1..31)/add(2^binomial(n,2)*x^n/n!,n=0..31);
%t A053549 f[x_, lim_] := Sum[2^Binomial[n, 2]*x^n/(n - 1)!, {n, 1, lim}] / Sum[2^Binomial[n, 2]*x^n/n!, {n, 0, lim}]; nn = 15; Range[0, nn]! CoefficientList[Series[f[x, nn], {x, 0, nn}], x] (* _T. D. Noe_, Oct 21 2011 *)
%o A053549 (PARI) q=30; my(x='x+O('x^20)); concat([0], Vec(serlaplace( sum(j=1,q, 2^binomial(j, 2)*x^j/(j-1)!)/(sum(k=0,q,2^binomial(k, 2)*x^k/k!)) ))) \\ _G. C. Greubel_, May 16 2019
%o A053549 (Magma) q:=30; m:=20; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[2^Binomial(j, 2)*x^j/Factorial(j-1): j in [1..q]])/(&+[2^Binomial(k, 2)*x^k/Factorial(k):k in [0..q]]) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, May 16 2019
%o A053549 (Sage) q=30; m = 20; T = taylor(sum(2^binomial(j, 2)*x^j/factorial(j-1) for j in (1..q))/(sum(2^binomial(k, 2)*x^k/factorial(k) for k in (0..q))), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, May 16 2019
%Y A053549 Cf. A006125.
%K A053549 nonn
%O A053549 0,3
%A A053549 _N. J. A. Sloane_, Jan 16 2000