A272905 Number of connected 4-regular (or quartic) labeled graphs with n nodes.

0, 0, 0, 0, 1, 15, 465, 19355, 1024380, 66462480, 5188446900, 480413448900, 52113339432000, 6551243302804200, 945313572845842200, 155243683741953807000, 28797215441570535960000, 5993001571565164940784000, 1390759438984816084192008000

Offset: 1

Author

Catherine Greenhill, May 09 2016

Keywords

• nonn

Comments

The e.g.f. for this sequence is the logarithm of the e.g.f. for the sequence of all 4-regular labeled graphs on n nodes (see A005815), using Wilf's exponential formula.

Examples

			The triangle of 4-valend labeled graphs with n>=1 nodes and 1<=k<=n components (row sums A005815) starts:
  0;
  0,0;
  0,0,0;
  0,0,0,0;
  1,0,0,0,0;
  15,0,0,0,0,0;
  465,0,0,0,0,0,0;
  19355,0,0,0,0,0,0,0;
  1024380,0,0,0,0,0,0,0,0;
  66462480,126,0,0,0,0,0,0,0,0;
  5188446900,6930,0,0,0,0,0,0,0,0,0;
  480413448900,472230,0,0,0,0,0,0,0,0,0,0;
  52113339432000,36878985,0,0,0,0,0,0,0,0,0,0,0;
  6551243302804200,3293696835,0,0,0,0,0,0,0,0,0,0,0,0;
  945313572845842200,334407638565,126126,0,0,0,0,0,0,0,0,0,0,0,0;
  155243683741953807000,38506555125675,15135120,0,0,0,0,0,0,0,0,0,0,0,0,0; - _R. J. Mathar_, Apr 29 2019
		

References

• H. S. Wilf, generatingfunctionology (2nd edn.), Academic Press, 1994, Corollary 3.4.1, page 81.

Links

• Catherine Greenhill, Table of n, a(n) for n = 1..100
• Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 13.

Crossrefs

Column k=4 of A324163.

See A005815 for not-necessarily-connected labeled 4-regular graphs.

Programs

Maple

egf := log((1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4], [], -12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16)));
ser := convert(series(egf, x=0, 40), polynom):
seq(coeff(ser, x, i)*i!, i=0..degree(ser));

Mathematica

g[x_] := Log[(Exp[x*(6-x^2)/8/(2+x)]* HypergeometricPFQ[{1/4, 3/4}, {}, ((12 (1-x) * x *(2 + x))/(x^3 + 2*x^2 - 6*x - 6)^2)])/ Sqrt[1 + x - x^2/3 - x^3/6]]; Rest[ CoefficientList[ Series[g[x], {x, 0, 30}], x]* Range[0, 30]!] (* Giovanni Resta, May 09 2016 *)

Formula

E.g.f.: log((1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16))).