A053549 Number of labeled rooted connected graphs.

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

Offset

0,3

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

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).

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 *)

Prog

(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
(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
(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

Crossrefs

Cf. A006125.

Sequence in context: A000795 A085628 A177777 * A139383 A216351 A365863

Adjacent sequences: A053546 A053547 A053548 * A053550 A053551 A053552

Keyword

nonn

Author

N. J. A. Sloane, Jan 16 2000

Status

approved