%I M4991 #69 Aug 28 2024 16:37:21
%S 1,0,0,0,0,1,15,465,19355,1024380,66462606,5188453830,480413921130,
%T 52113376310985,6551246596501035,945313907253606891,
%U 155243722248524067795,28797220460586826422720
%N Number of 4-valent labeled graphs with n nodes.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A005815/b005815.txt">Table of n, a(n) for n = 0..260</a> (first 100 terms from Vincenzo Librandi)
%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. 9.
%H I. P. Goulden, D. M. Jackson, and J. W. Reilly, <a href="https://web.archive.org/web/20220129133942/https://uwaterloo.ca/math/sites/ca.math/files/uploads/files/gjsiad1983.pdf">The Hammond series of a symmetric function and its application to P-recursiveness</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
%H Atabey Kaygun, <a href="https://arxiv.org/abs/2101.02299">Enumerating Labeled Graphs that Realize a Fixed Degree Sequence</a>, arXiv:2101.02299 [math.CO], 2021.
%H Vaclav Kotesovec, <a href="/A005815/a005815.txt">Recurrence (of order 10)</a>
%H R. C. Read and N. C. Wormald, <a href="http://dx.doi.org/10.1002/jgt.3190040208">Number of labeled 4-regular graphs</a>, J. Graph Theory, 4 (1980), 203-212.
%F From _Vladeta Jovovic_, Mar 26 2001: (Start)
%F E.g.f. f(x) = Sum_{n >= 0} a(n)*x^n/(n)! satisfies the differential equation 16*x^2*(x - 1)^2*(x + 2)^2*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)*(d^2/dx^2)y(x) - 4*(x^13 + 4*x^12 - 16*x^10 - 10*x^9 - 36*x^8 - 220*x^7 - 348*x^6 - 48*x^5 + 200*x^4 - 336*x^3 - 240*x^2 + 416*x - 96)*(d/dx)y(x) - x^4*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)^2*y(x) = 0.
%F Recurrence: a(n) = - 1/384*(( - 256*n^2 - 896*n + 1152)*a(n - 1) + (768*n^3 - 3648*n^2 + 5568*n - 2688)*a(n - 2) + ( - 192*n^4 + 3264*n^3 - 14784*n^2 + 24384*n - 12672)*a(n - 3) + (224*n^6 - 4512*n^5 + 36304*n^4 - 148160*n^3 + 320016*n^2 - 341728*n + 137856)*a(n - 5) + ( - 640*n^5 + 8800*n^4 - 46400*n^3 + 116000*n^2 - 135360*n + 57600)*a(n - 4) + ( - 24*n^10 + 1320*n^9 - 31680*n^8 + 435600*n^7 - 3786552*n^6 + 21649320*n^5 - 82006320*n^4 + 201828000*n^3 - 306085824*n^2 + 255087360*n - 87091200)*a(n - 11) + (64*n^10 - 3480*n^9 + 82692*n^8 - 1127232*n^7 + 9726024*n^6 - 55255032*n^5 + 208179908*n^4 - 510068208*n^3 + 770738352*n^2 - 640484928*n + 218211840)*a(n - 9) + (16*n^11 - 992*n^10 + 27256*n^9 - 437160*n^8 + 4536288*n^7 - 31876656*n^6 + 154182488*n^5 - 510784360*n^4 + 1128552896*n^3 - 1570313952*n^2 + 1223830656*n - 397716480)*a(n - 10) + ( - 128*n^8 + 5488*n^7 - 94576*n^6 + 864976*n^5 - 4606672*n^4 + 14604352*n^3 - 26753984*n^2 + 25611264*n - 9630720)*a(n - 7) + (16*n^9 - 576*n^8 + 8704*n^7 - 71680*n^6 + 348880*n^5 - 1013824*n^4 + 1673376*n^3 - 1333120*n^2 + 226944*n + 161280)*a(n - 8) + (128*n^7 - 2192*n^6 + 12048*n^5 - 8240*n^4 - 151248*n^3 + 565312*n^2 - 765248*n + 349440)*a(n - 6) + ( - 4*n^13 + 364*n^12 - 14924*n^11 + 364364*n^10 - 5897892*n^9 + 66678612*n^8 - 540145892*n^7 + 3163772612*n^6 - 13344475144*n^5 + 39830815024*n^4 - 81255012384*n^3 + 106386868224*n^2 - 79211036160*n + 24908083200)*a(n - 14) + ( - 4*n^13 + 360*n^12 - 14612*n^11 + 353496*n^10 - 5674812*n^9 + 63680760*n^8 - 512439356*n^7 + 2983811688*n^6 - 12520194544*n^5 + 37201987680*n^4 - 75598952832*n^3 + 98660630016*n^2 - 73265264640*n + 22992076800)*a(n - 13) + ( - 16*n^12 + 1244*n^11 - 43208*n^10 + 884620*n^9 - 11860728*n^8 + 109396452*n^7 - 709293464*n^6 + 3243764260*n^5 - 10331326456*n^4 + 22203205904*n^3 - 30301280928*n^2 + 23300910720*n - 7504358400)*a(n - 12) + ( - n^14 + 105*n^13 - 5005*n^12 + 143325*n^11 - 2749747*n^10 + 37312275*n^9 - 368411615*n^8 + 2681453775*n^7 - 14409322928*n^6 + 56663366760*n^5 - 159721605680*n^4 + 310989260400*n^3 - 392156797824*n^2 + 283465647360*n - 87178291200)*a(n - 15)). (End)
%F a(n) = Sum_{d=0..floor(n/2), c=0..floor(n/2-d), b=0..(n-2c-2d), f=0..(n-2c-2d-b), k=0..min(n-b-2c-2d-f, 2n-2f-2b-3c-4d), j=0..floor(k/2+f)} ((-1)^(k+2f-j+d)*n!*(k+2f)!(2(2n-k-2f-2b-3c-4d))!) / (2^(5n-2k-2f-3b-8c-7d) * 3^(n-b-c-2d-k-f)*(2n-k-2f-2b-3c-4d)!*(k+2f-2j)!*j!*b!*c!*d!*k!*f!*(n-b-2c-2d-k-f)!). - _Shanzhen Gao_, Jun 05 2009
%F E.g.f.: (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)). - _Mark van Hoeij_, Nov 07 2011
%F a(n) ~ n^(2*n) * 2^(n+1/2) / (3^n * exp(2*n+15/4)). - _Vaclav Kotesovec_, Mar 11 2014
%p egf := (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)); # _Mark van Hoeij_, Nov 07 2011
%t max = 17; f[x_] := HypergeometricPFQ[{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)]/(1 + x - x^2/3 - x^3/6)^ (1/2); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, Jun 19 2012, from e.g.f. *)
%Y Cf. A005814, A002829, A005816, A272905 (connected). A diagonal of A059441.
%K nonn,nice,easy
%O 0,7
%A _Simon Plouffe_
%E More terms from _Vladeta Jovovic_, Mar 26 2001