Relations between A241885/A242225, A222411/A222412, and A350194/A350154.

David Broadhurst, 
Open University, UK
(Email to N. J. A. Sloane, Apr 23 2022)

Theorem:
(1) A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n))
(2) A241885(n)/A242225(n) = n!*A350194(n)/(A350154(n)*(2*n+1))

Proofs:
A241885(n)/A242225(n) is the coefficient of x^n/n! in
g1(x) = (x/(exp(x)-1))^(1/2).
A222411(n)/A222412(n) is the coefficient of x^n in
g2(x) = (x/(exp(x)-1))^(3/2)*exp(x/2).
A350194(n)/A350154(n) is the coefficient of x^(2*n+1) in
g3(x) = arccos(exp(-x^2/2)).
Differentiation of g1 and g3 shows that
g2(-x) = g1(x) - 2*x*g1'(x), which proves (1), and that
g3'(x) = g1(x^2), which proves (2).

Here is Pari code, up to n = 300:

{nmax=300;v = Vec(sqrt(x/(exp(x+x^2*O(x^nmax))-1))-1);}
{A241885(n) = if(n<1, 1, numerator(v[n]*n!));}
{A242225(n) = if(n<1, 1, denominator(v[n]*n!));}
{A222411(n) = if(n<1, 1, numerator(v[n]*(-1)^n*(1-2*n)));}
{A222412(n) = if(n<1, 1, denominator(v[n]*(-1)^n*(1-2*n)));}
{A350194(n) = if(n<1, 1, numerator(v[n]/(2*n+1)));}
{A350154(n) = if(n<1, 1, denominator(v[n]/(2*n+1)));}

Remark:
Relations between fractional sequences may be hidden in OEIS,
which can list only their integer numerators and denominators.
Yet these may be revealed by large prime numerators. The results above 
were discovered by using the prime 131301607, which occurred in my work.

The search
https://oeis.org/search?q=131301607 revealed that
A241885(14) = -A222411(14) = A350194(14) = 131301607.
which led to the Theorem.

Best wishes,

David Broadhurst
23 April 2022