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%I #30 Apr 21 2023 02:49:04
%S 1,1,8,96,2112,68160,3087360,185633280,14301020160,1372232171520,
%T 160390869811200,22426206024499200,3695148753459609600,
%U 708443854690399027200,156340439420689081958400,39342248735234589720576000,11197266840049016358567936000
%N Sum_{n >= 0} a(n) * x^n / n!^2 = exp(-2*x)/(1-x)^3.
%C As described in the Stanley reference, this sequence gives the expectation of the fourth moment of a random sign matrix (a matrix whose entries are independently set equal to -1 or 1 with equal probability) of size n. For large n, a(n) is asymptotic to (n!)^2*(n^2+7n+10)/(2e^2). - Kevin P. Costello (kcostell(AT)gmail.com), Oct 22 2007
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).
%D G. Szekeres, The average value of skew Hadamard matrices, Proceedings of the First Australian Conference on Combinatorial Mathematics (Univ. Newcastle, Newcastle, 1972), pp. 55--59. Univ. of Newcastle Res. Associates, Newcastle, 1972. MR0349708 (50 #2201). This is S_4(n).
%H Zelin Lv and Aaron Potechin, <a href="https://arxiv.org/abs/2206.11356">The Sixth Moment of Random Determinants</a>, arXiv:2206.11356 [math.CO], 2022. See Table 1 p. 3.
%H H. Nyquist, S. O. Rice, and J. Riordan, <a href="https://www.jstor.org/stable/43634123">The distribution of random determinants</a>, Quarterly of Applied Mathematics, 12(2):97-104, 1954.
%F a(n) = (n!)^2*A209429(n)/A209430(n). [Szekeres]
%F a(n) = n! * A052124(n). - _Sean A. Irvine_, Oct 25 2021
%o (PARI) my(x='x+O('x^30), v = Vec(serlaplace( exp(-2*x)/(1-x)^3))); vector(#v, k, v[k]*(k-1)!) \\ _Michel Marcus_, Oct 25 2021
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o def A052127(n): return int((n+5)*(n+2)*factorial(n)**2*sum(Fraction((-1 if k&1 else 1)*(k+3)<<k+2,factorial(k+5)) for k in range(n+1))) # _Chai Wah Wu_, Apr 20 2023
%Y Cf. A052124, A209429, A209430.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Jan 23 2000
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