OFFSET
0,3
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.
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Zelin Lv, Oct 03 2022
EXTENSIONS
a(0)=1 prepended and some terms corrected by Alois P. Heinz, Apr 19 2023
STATUS
approved