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A005814 Number of 3-regular (trivalent) labeled graphs on 2n vertices with multiple edges and loops allowed.
(Formerly M2168)
7
1, 2, 47, 4720, 1256395, 699971370, 706862729265, 1173744972139740, 2987338986043236825, 11052457379522093985450, 57035105822280129537568575, 397137564714721907350936061400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 175, (7.5.12).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60--66. MR0819706 (87k:05093)

I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.

K. Phillips, R functions to symbolically compute the central moments of the multivariate normal distribution, Journal of Statistical Software, Feb 2010.

K. Phillips, symmoments R package

Kem Phillips, Proof for multivariate normal moments

FORMULA

From Vladeta Jovovic, Mar 25 2001: (Start)

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.

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.

(End)

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

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]

Equation (7.5.13) in Harary and Palmer gives asymptotic formula.

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

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

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

EXAMPLE

a(1)=2: {(1,1), (1,2), (2,2)}, {(1,2), (1,2), (1,2)}.

MATHEMATICA

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-Fran├žois Alcover, Nov 29 2011 *)

CROSSREFS

Cf. A002829, A002135.

Sequence in context: A119776 A087265 A079307 * A177190 A087259 A195876

Adjacent sequences:  A005811 A005812 A005813 * A005815 A005816 A005817

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

More terms from Vladeta Jovovic, Mar 25 2001

Edited by N. J. A. Sloane, Apr 19 2007

STATUS

approved

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Last modified February 21 04:43 EST 2018. Contains 299389 sequences. (Running on oeis4.)