A272905 - OEIS

%I #23 Aug 28 2024 12:40:18

%S 0,0,0,0,1,15,465,19355,1024380,66462480,5188446900,480413448900,

%T 52113339432000,6551243302804200,945313572845842200,

%U 155243683741953807000,28797215441570535960000,5993001571565164940784000,1390759438984816084192008000

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

%C 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.

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

%H Catherine Greenhill, <a href="/A272905/b272905.txt">Table of n, a(n) for n = 1..100</a>

%H Élie de Panafieu, <a href="https://arxiv.org/abs/2408.12459">Asymptotic expansion of regular and connected regular graphs</a>, arXiv:2408.12459 [math.CO], 2024. See p. 13.

%F 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))).

%e The triangle of 4-valend labeled graphs with n>=1 nodes and 1<=k<=n components (row sums A005815) starts

%e 0;

%e 0,0;

%e 0,0,0;

%e 0,0,0,0;

%e 1,0,0,0,0;

%e 15,0,0,0,0,0;

%e 465,0,0,0,0,0,0;

%e 19355,0,0,0,0,0,0,0;

%e 1024380,0,0,0,0,0,0,0,0;

%e 66462480,126,0,0,0,0,0,0,0,0;

%e 5188446900,6930,0,0,0,0,0,0,0,0,0;

%e 480413448900,472230,0,0,0,0,0,0,0,0,0,0;

%e 52113339432000,36878985,0,0,0,0,0,0,0,0,0,0,0;

%e 6551243302804200,3293696835,0,0,0,0,0,0,0,0,0,0,0,0;

%e 945313572845842200,334407638565,126126,0,0,0,0,0,0,0,0,0,0,0,0;

%e 155243683741953807000,38506555125675,15135120,0,0,0,0,0,0,0,0,0,0,0,0,0; - _R. J. Mathar_, Apr 29 2019

%p 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)));

%p ser := convert(series(egf, x=0, 40), polynom):

%p seq(coeff(ser, x, i)*i!, i=0..degree(ser));

%t 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 *)

%Y Column k=4 of A324163.

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

%K nonn

%O 1,6

%A _Catherine Greenhill_, May 09 2016