A209429 -id:A209429 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 23 2000

nonn

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

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).

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.

Cf. A052124, A209429, A209430.

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

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

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

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

A209430 Denominator of l(n), where l(1)=1, l(2)=2, l(n)=l(n-1)+2*l(n-2)/n.

Original entry on oeis.org

1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, 155925, 467775, 6081075, 42567525, 58046625, 638512875, 10854718875, 97692469875, 1856156927625, 9280784638125, 194896477400625, 2143861251406875, 49308808782358125, 147926426347074375, 217538862275109375, 4370553505709015625

Offset: 1

N. J. A. Sloane, Mar 22 2012

			1, 2, 8/3, 11/3, 71/15, 268/45, 2302/315, 2771/315, 29543/2835, 172654/14175, 2194624/155925, 7533469/467775, 111102841/6081075, 875654984/42567525, ...

Szekeres, G. 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)

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A209429, A052127.

Denominator[RecurrenceTable[{a[1]==1,a[2]==2,a[n]==a[n-1]+(2a[n-2])/n},a,{n,30}]] (* Harvey P. Dale, Mar 30 2014 *)

cp's OEIS Frontend

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

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