A357571 The sixth moment of an n X n random +-1 matrix.

1, 1, 32, 1536, 282624, 66846720, 27053752320, 16104538275840, 13681567224299520, 15874223643851489280, 24412997036693834956800, 48514602066025722465484800, 121994703799547846503012761600, 381343447691461317926230740172800, 1459468400650603118890910517244723200

Offset: 0

Author

Zelin Lv, Oct 03 2022

Keywords

• nonn

Comments

This sequence gives the expectation of the sixth moment of a random sign matrix (a matrix whose entries are independently set equal to -1 or 1 with equal probability) of size n X n.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..180
• Dominik Beck, Zelin Lv and Aaron Potechin, The Sixth Moment of Random Determinants, arXiv:2206.11356 [math.CO], 2022-2023. See Table 1 p. 4.

Crossrefs

Cf. A052127.

Programs

PARI

f6(n,m4,m6)=sum(j=0,n, binomial(n,j)*sum(a=0,j, binomial(j,a)*(m6-15)^a*(m4-3)^(j-a)*D(n,a,j-a)))
D(n,a,b)=prod(j=0,a+b-1,n-j)*sum(i=0,b, binomial(b,i)*C(i)*H(n,b-i,a,b))*P(n-a-b)
P(n)=n!*(n+2)!*(n+4)!/48
C(n)=if(n<2, n==0, (n-1)*(C(n-1)+15*C(n-2)))
H(n,j,a,b)=sum(x=1,j,binomial(j-1,x-1)*j!/x!*prod(y=0,x-1, 3*(n-a-b)-y))
\\ Charles R Greathouse IV, Oct 03 2022

PARI

a(n)={(n!)^2 * sum(j=0, n, sum(i=0, j, ((1+i)*(2+i)*(4+i)! / (48*(n-j)!)) * binomial(14+j+2*i,j-i) * (16)^(n-j) * (-2)^(j-i) ))} \\ Andrew Howroyd, Mar 16 2023

Python

from fractions import Fraction
from math import factorial, comb
def A357571(n): return int(factorial(n)**2*sum(Fraction(1<<(n-j<<2),3*factorial(n-j))*sum((1+i)*(2+i)*factorial(4+i)*comb(14+j+(i<<1),j-i)*(-1 if (j-i)&1 else 1)<<(j-i) for i in range(j+1)) for j in range(n+1)))>>4 # Chai Wah Wu, Apr 20 2023

Formula

a(n) = (n!)^2 * Sum_{j=0..n} Sum_{i=0..j} ((1+i)*(2+i)*(4+i)! / (48*(n-j)!)) * binomial(14+j+2*i,j-i) * (m_6-15*m_4+30)^(n-j) * (m_4-3)^(j-i) where m_6 = m_4 = 1.

G.f.: A(t) = (e^(t*(m_6 - 15*m_4 + 30)) / (48*(1 + 3*t - m_4*t)^15)) * Sum_{i>=0} ((1+i)*(2+i)*(4+i)! * t^i / (1 + 3*t - m_4*t)^(3*i)) where m_6 = m_4 = 1.

Extensions

a(0)=1 prepended and some terms corrected by Alois P. Heinz, Apr 19 2023