A053549 Number of labeled rooted connected graphs.

0, 1, 2, 12, 152, 3640, 160224, 13063792, 2012388736, 596666619648, 344964885948160, 392058233038486784, 880255154481199466496, 3916538634445633156373504, 34603083354426212294072477696, 607915214065957203519146330173440

Offset: 0

Author

N. J. A. Sloane, Jan 16 2000

Keywords

• nonn

References

• F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 10, R_p.
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.20, G(x).

Links

• T. D. Noe, Table of n, a(n) for n=0..50

Crossrefs

Cf. A006125.

Programs

Magma

q:=30; m:=20; R:=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

Maple

add(2^binomial(n,2)*x^n/(n-1)!,n=1..31)/add(2^binomial(n,2)*x^n/n!,n=0..31);

Mathematica

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 *)

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

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

Formula

E.g.f.: (Sum_{n>1} 2^binomial(n, 2)*x^n/(n-1)!)/(Sum_{n>=0} 2^binomial(n, 2)*x^n/n!).

a(n) = n * A001187(n).