A052127 Sum_{n >= 0} a(n) * x^n / n!^2 = exp(-2*x)/(1-x)^3.

1, 1, 8, 96, 2112, 68160, 3087360, 185633280, 14301020160, 1372232171520, 160390869811200, 22426206024499200, 3695148753459609600, 708443854690399027200, 156340439420689081958400, 39342248735234589720576000, 11197266840049016358567936000

Offset: 0

Author

N. J. A. Sloane, Jan 23 2000

Keywords

• nonn

Comments

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

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.64(b).
• 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).

Links

• Zelin Lv and Aaron Potechin, The Sixth Moment of Random Determinants, arXiv:2206.11356 [math.CO], 2022. See Table 1 p. 3.
• H. Nyquist, S. O. Rice, and J. Riordan, The distribution of random determinants, Quarterly of Applied Mathematics, 12(2):97-104, 1954.

Crossrefs

Cf. A052124, A209429, A209430.

Programs

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

Python

from math import factorial
from fractions import Fraction
def A052127(n): return int((n+5)*(n+2)*factorial(n)**2*sum(Fraction((-1 if k&1 else 1)*(k+3)<Chai Wah Wu, Apr 20 2023

Formula

a(n) = (n!)^2*A209429(n)/A209430(n). [Szekeres]

a(n) = n! * A052124(n). - Sean A. Irvine, Oct 25 2021