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 A272905 Number of connected 4-regular (or quartic) labeled graphs with n nodes. 3
 0, 0, 0, 0, 1, 15, 465, 19355, 1024380, 66462480, 5188446900, 480413448900, 52113339432000, 6551243302804200, 945313572845842200, 155243683741953807000, 28797215441570535960000, 5993001571565164940784000, 1390759438984816084192008000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 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. 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. 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))). EXAMPLE 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 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 *) CROSSREFS Column k=4 of A324163. See A005815 for not-necessarily-connected labeled 4-regular graphs. Sequence in context: A320097 A177080 A306609 * A005815 A120600 A279922 Adjacent sequences: A272902 A272903 A272904 * A272906 A272907 A272908 KEYWORD nonn,changed AUTHOR Catherine Greenhill, May 09 2016 STATUS approved

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Last modified September 10 02:52 EDT 2024. Contains 375770 sequences. (Running on oeis4.)