|
%I M2168 #68 Mar 25 2020 12:44:04
%S 1,2,47,4720,1256395,699971370,706862729265,1173744972139740,
%T 2987338986043236825,11052457379522093985450,
%U 57035105822280129537568575,397137564714721907350936061400
%N Number of 3-regular (trivalent) labeled graphs on 2n vertices with multiple edges and loops allowed.
%C a(n) is the number of representations required for the symbolic central moments of order 3 for the multivariate normal distribution, that is, E[X1^3 X2^3 .. Xn^3|mu=0, Sigma], where n is even. These representations are the upper-triangular, positive integer matrices for which for each i, the sum of the i-th row and i-th column equals 3, the power of each component. See Phillips links below. - _Kem Phillips_, Aug 18 2014
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 175, (7.5.12).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005814/b005814.txt">Table of n, a(n) for n = 0..100</a>
%H I. P. Goulden and D. M. Jackson, <a href="http://dx.doi.org/10.1137/0607007">Labelled graphs with small vertex degrees and P-recursiveness</a>, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093)
%H I. P. Goulden, D. M. Jackson, and J. W. Reilly, <a href="http://dx.doi.org/10.1137/0604019">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 K. Phillips, <a href="http://dx.doi.org/10.18637/jss.v033.c01">R functions to symbolically compute the central moments of the multivariate normal distribution</a>, Journal of Statistical Software, Feb 2010.
%H K. Phillips, <a href="http://CRAN.R-project.org/package=symmoments">symmoments R package</a>
%H Kem Phillips, <a href="/A005814/a005814_1.pdf">Proof for multivariate normal moments</a>
%F From _Vladeta Jovovic_, Mar 25 2001: (Start)
%F E.g.f. f(x) = Sum_{n>=0} a(2 * n) * x^n/(2 * n)! satisfies the differential equation 6 * x^2 * (x^2 - 2 * x - 2) * (d^2/dx^2)f(x) - (x^5 - 6 * x^4 + 6 * x^3 + 24 * x^2 + 16 * x - 8) * (d/dx)f(x) + (1/6) * (x^5 - 10 * x^4 + 24 * x^3 - 4 * x^2 - 44 * x - 48) * f(x) = 0.
%F Recurrence: a(2 * n) = (2 * n)!/n! * v(n) where 48 * v(n) + (-72 * n^2 + 120 * n - 96) * v(n - 1) + (-72 * n^3 + 288 * n^2 - 404 * n + 188) * v(n - 2) + (36 * n^4 - 396 * n^3 + 1472 * n^2 - 2184 * n + 1072) * v(n - 3) + (36 * n^4 - 336 * n^3 + 1116 * n^2 - 1536 * n + 720) * v(n - 4) + (-6 * n^5 + 80 * n^4 - 410 * n^3 + 1000 * n^2 - 1144 * n + 480) * v(n - 5) + (n^5 - 15 * n^4 + 85 * n^3 - 225 * n^2 + 274 * n - 120) * v(n - 6) = 0.
%F (End)
%F Linear recurrence satisfied by a(n): {a(0) = 1, a(1) = 2, a(2) = 47, a(3) = 4720, a(4) = 1256395, a(5) = 699971370, and (4989600 + 5718768*n^7 + 1045440*n^8 + 123200*n^9 + 8448*n^10 + 256*n^11 + 30135960*n + 75458988*n^2 + 105258076*n^3 + 91991460*n^4 + 53358140*n^5 + 21100464*n^6)*a(n) + (-39916800 - 1756320*n^7 - 198720*n^8 - 13120*n^9 - 384*n^10 - 136306080*n - 205327944*n^2 - 179845580*n^3 - 101513280*n^4 - 38608500*n^5 - 10026072*n^6)*a(n + 1) + (19958400 + 17664*n^7 + 576*n^8 + 44868240*n + 43664892*n^2 + 24024336*n^3 + 8173284*n^4 + 1760640*n^5 + 234528*n^6)*a(n + 2) + (720720 + 144*n^7 + 1819364*n + 1758924*n^2 + 883226*n^3 + 254070*n^4 + 42356*n^5 + 3816*n^6)*a(n + 3) + (-183645 - 191119*n - 79608*n^2 - 16586*n^3 - 1728*n^4 - 72*n^5)*a(n + 4) + (-2706 - 1515*n - 285*n^2 - 18*n^3)*a(n + 5) + 3*a(n + 6)}. - _Marni Mishna_, Jun 17 2005
%F Linear differential equation satisfied by F(t)=Sum a(n) t^n/(2n)!: {F(0) = 1, - 3*t*(10*t^2 + 9*t^6 + 18*t^4 - 8 + t^10 - 6*t^8)*( - 2 - 2*t^2 + t^4)*(d/dt)F(t) + 9*t^4*( - 2 - 2*t^2 + t^4)^2*(d^2/dt^2)F(t) + t^2*(-2 - 2*t^2 + t^4)*(24*t^6 - 10*t^8 - 4*t^4 - 44*t^2 + t^10 - 48)*F(t)}. - _Marni Mishna_, Jun 17 2005 [Probably this defines A005814? - _N. J. A. Sloane_]
%F Equation (7.5.13) in Harary and Palmer gives asymptotic formula.
%F Asymptotic formula (7.5.13) exp(-2)*(6*n)!/(288^n*(3*n)!) by Harary and Palmer from this reference is for sequence A002829. - _Vaclav Kotesovec_, Mar 11 2014
%F Asymptotic for A005814 is: a(n) ~ exp(2) * (6*n)! / (288^n * (3*n)!), or a(n) ~ sqrt(2) * 6^n * n^(3*n) / exp(3*n-2). - _Vaclav Kotesovec_, Mar 11 2014
%F Recurrence (of order 4): 3*a(n) = 9*(n-1)*n*(2*n-1)*a(n-1) + (n-1)*(2*n-3)*(2*n-1)*(12*n-1)*a(n-2) - 2*(n-2)*n*(2*n-5)*(2*n-3)*(2*n-1)*(3*n-2)*a(n-3) + 2*(n-3)*(n-1)*n*(2*n-7)*(2*n-5)*(2*n-3)*(2*n-1)*a(n-4). - _Vaclav Kotesovec_, Mar 11 2014
%e a(1)=2: {(1,1), (1,2), (2,2)}, {(1,2), (1,2), (1,2)}.
%t max = 11; f[x_] := Sum[a[2n]*(x^n/(2n)!), {n, 0, max}]; a[0] = 1; coes = CoefficientList[ 6x^2*(x^2 - 2x - 2)* f''[x] - (x^5 - 6x^4 + 6x^3 + 24x^2 + 16x - 8)*f'[x] + 1/6*(x^5 - 10x^4 + 24x^3 - 4x^2 - 44x - 48)*f[x], x]; Table[a[2 n], {n, 0, max}] /. Solve[Thread[coes[[1 ;; max]] == 0]][[1]](* _Jean-François Alcover_, Nov 29 2011 *)
%Y Even bisection of column k=3 of A333467.
%Y Cf. A002829, A002135.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _Vladeta Jovovic_, Mar 25 2001
%E Edited by _N. J. A. Sloane_, Apr 19 2007
| 