from fractions import Fraction
def rayleigh(nu, n):
nu = Fraction(nu)
if nu < 0 or n < 1:
raise ValueError("Require nu >= 0 and integer n >= 1")
s = [Fraction(0), Fraction(1, 4) / (nu + 1)]
for m in range(2, n + 1):
s.append(sum(s[k] * s[m-k] for k in range(1, m)) / (nu + m))
return s[n]
rayleigh("1/2", 12)
All 96 values were obtained in exact rational arithmetic by coefficient comparison in the logarithmic derivative of the normalized Bessel series [6] and its canonical product [5]. No decimal approximation or truncated root sum enters the values.
A separate implementation using Python fractions agrees with the convolution recurrence [1] on every entry. The first three moment formulas agree at all eight orders. Every half-order specialization agrees with independently computed Bernoulli numbers and with the even-zeta formula evaluated at 120 decimal digits. Positivity, strict decrease and strict log convexity hold throughout the stored range. A supplementary numerical sum over 32 positive roots per order, with an explicit upper bound for the infinite tail, agrees with all 96 values. That check uses numerically indexed roots and does not certify 100 digits; the exact rational computation supplies the stated rigour. The attached checks include known nonzero controls and a deliberately perturbed value, and were repeated on the values read back from the database.