A382257 a(n) is the numerator of tanh(Sum_{k=1..n-1} artanh(k/n)), where artanh is the inverse hyperbolic tangent function.

0, 1, 9, 17, 125, 461, 1715, 3217, 24309, 92377, 352715, 1352077, 5200299, 20058299, 77558759, 150270097, 1166803109, 4537567649, 17672631899, 68923264409, 269128937219, 1052049481859, 4116715363799, 16123801841549, 63205303218875, 247959266474051, 973469712824055, 3824345300380219, 15033633249770519

Offset

1,3

Comments

The value of tanh(...) is always a rational number thanks to the relation tanh(x+y) = (tanh x + tanh y)/(1 + (tanh x)*tanh y).

So actually the set of fractions {1/n, ..., (n-1)/n} is "summed up" using the operator x (+) y := (x + y)/(1 + x*y).

By Wolstenholme's theorem; if p > 3 is prime, then p^3 divides a(p). - Thomas Ordowski, Apr 27 2025

Links

Table of n, a(n) for n=1..29.

Wikipedia, Area function (inverse hyperbolic function)

Formula

a(n) = (binomial(2n-1, n-1) - 1)/2 if n = 2^m or a(n) = binomial(2n-1, n-1) - 1 = A010763(n-1) otherwise, since tanh(Sum_{k=1..n-1} artanh(k/n)) = (binomial(2n-1, n-1) - 1)/(binomial(2n-1, n-1) + 1) reduced. - Thomas Ordowski, Apr 27 2025

Example

For n=2, tanh(artanh(1/2)) = 1/2, so a(2) = numerator(1/2) = 1.
For n=3, tanh(artanh(1/3) + artanh(2/3)) = (1/3 + 2/3) / (1 + 1/3 * 2/3) = 9/11, so a(3) = 9.
Numerators of 0, 1/2, 9/11, 17/18, 125/127, 461/463, 1715/1717, 3217/3218, ...

Prog

(Python)
from sympy import S, expand_trig as ET
tanh, artanh = S("tanh, artanh")
def A382257_test(n): # for illustration only -- slow for n >= 19
    n=S(n); return ET(tanh(sum(artanh(k/n) for k in range(1, n)))).numerator
def A382257(n):
    s=0; i=S.One/n
    for k in range(1, n): s = (s + i*k)/(1 + s*k*i)
    return s.numerator
(Python)
from functools import reduce
from fractions import Fraction
def A382257(n): return reduce(lambda x, y: Fraction(x+y, 1+x*y), (Fraction(i, n) for i in range(1, n)), 0).numerator # Chai Wah Wu, Apr 23 2025
(PARI) A382257(n)=my(s=0); for(i=1, n-1, s=(s*n+i)/(n+s*i)); numerator(s);

Crossrefs

Cf. A001700, A010763, A034602, A383431 (denominators).

Sequence in context: A049440 A177200 A177166 * A073221 A110462 A132375

Adjacent sequences: A382254 A382255 A382256 * A382258 A382259 A382260

Keyword

nonn,frac

Author

M. F. Hasler, Apr 15 2025

Status

approved