cp's OEIS Frontend

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

%I A272905 #26 May 20 2025 03:12:32
%S A272905 0,0,0,0,1,15,465,19355,1024380,66462480,5188446900,480413448900,
%T A272905 52113339432000,6551243302804200,945313572845842200,
%U A272905 155243683741953807000,28797215441570535960000,5993001571565164940784000,1390759438984816084192008000
%N A272905 Number of connected 4-regular (or quartic) labeled graphs with n nodes.
%C A272905 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 A272905 H. S. Wilf, generatingfunctionology (2nd edn.), Academic Press, 1994, Corollary 3.4.1, page 81.
%H A272905 Catherine Greenhill, <a href="/A272905/b272905.txt">Table of n, a(n) for n = 1..100</a>
%H A272905 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 A272905 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 A272905 The triangle of 4-valend labeled graphs with n>=1 nodes and 1<=k<=n components (row sums A005815) starts:
%e A272905   0;
%e A272905   0,0;
%e A272905   0,0,0;
%e A272905   0,0,0,0;
%e A272905   1,0,0,0,0;
%e A272905   15,0,0,0,0,0;
%e A272905   465,0,0,0,0,0,0;
%e A272905   19355,0,0,0,0,0,0,0;
%e A272905   1024380,0,0,0,0,0,0,0,0;
%e A272905   66462480,126,0,0,0,0,0,0,0,0;
%e A272905   5188446900,6930,0,0,0,0,0,0,0,0,0;
%e A272905   480413448900,472230,0,0,0,0,0,0,0,0,0,0;
%e A272905   52113339432000,36878985,0,0,0,0,0,0,0,0,0,0,0;
%e A272905   6551243302804200,3293696835,0,0,0,0,0,0,0,0,0,0,0,0;
%e A272905   945313572845842200,334407638565,126126,0,0,0,0,0,0,0,0,0,0,0,0;
%e A272905   155243683741953807000,38506555125675,15135120,0,0,0,0,0,0,0,0,0,0,0,0,0; - _R. J. Mathar_, Apr 29 2019
%p A272905 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 A272905 ser := convert(series(egf, x=0, 40), polynom):
%p A272905 seq(coeff(ser, x, i)*i!, i=0..degree(ser));
%t A272905 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 A272905 Column k=4 of A324163.
%Y A272905 See A005815 for not-necessarily-connected labeled 4-regular graphs.
%K A272905 nonn
%O A272905 1,6
%A A272905 _Catherine Greenhill_, May 09 2016