A362413 The second moment of an n X n symmetric random +-1 matrix.
1, 1, 2, 8, 44, 244, 1744, 13768, 127952, 1270736, 14384096, 172799296, 2306400832, 32442943168, 498547591424, 8031916728704, 139611091407104, 2533449773986048, 49133884886866432, 991341134236389376, 21218511171980205056, 471083434031674336256
Offset: 0
Keywords
References
- Zelin Lv, On The Moments of Random Determinants, Master Thesis, the University of Chicago.
- I. G. Zhurbenko, Moments of random determinants, Theory of Probability & Its Application, 13(4):682-686, 1968.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..448
Programs
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Maple
a:= n-> `if`(n=0, 1, q(n)*(n-1)!): p:= n-> `if`(n<3, 1, 3-irem(n, 2)): q:= proc(n) option remember; p(n)+add(p(n-i)*q(i)/i, i=1..n-1) end: seq(a(n), n=0..21); # Alois P. Heinz, Apr 19 2023 -
Mathematica
a[n_] := If[n == 0, 1, q[n]*(n-1)!]; p[n_] := If[n < 3, 1, 3-Mod[n, 2]]; q[n_] := q[n] = p[n] + Sum[p[n-i]*q[i]/i, {i, 1, n-1}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 24 2025, after Alois P. Heinz *) -
Python
from math import factorial from fractions import Fraction from functools import lru_cache @lru_cache(maxsize=None) def A362413(n): return int(((1 if n<=2 else (2 if n&1 else 3))+sum(Fraction((1 if n-i<=2 else (2 if n-i&1 else 3))*A362413(i),factorial(i)) for i in range(1,n)))*factorial(n-1)) if n else 1 # Chai Wah Wu, Apr 20 2023
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SageMath
x = LazyPowerSeriesRing(QQ, "x").gen() egf = exp(-x * (x + 1)) / sqrt((x + 1) * (1 - x)^5) [egf[n] * factorial(n) for n in range(22)] # Peter Luschny, Apr 20 2023
Formula
f^(sym)_2(n) = q(n) * (n-1)!, where
p(n) =
1, if n <= 2
2, if n >= 3 and n is odd
3, if n >= 4 and n is even
q(n) = p(n) + Sum_{i=1..n-1}(p(i) * q(n-i)) / (n-i).
E.g.f.: exp(-x*(x+1))/sqrt((x+1)*(1-x)^5). - Alois P. Heinz, Apr 19 2023
a(n) ~ 4*n^(n+2)/ (3*exp(n+2)). - Vaclav Kotesovec, Apr 20 2023
a(n) = (p(n) + Sum_{i=1..n-1} p(n-i) * a(i)/i! ) * (n-1)!. - Chai Wah Wu, Apr 20 2023
Comments