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-François Alcover, Nov 29 2011 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Mar 25 2001
Edited by N. J. A. Sloane, Apr 19 2007
STATUS
approved