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Searching the numerators and denominators in the Online encyclopedia of integer sequences (OEIS):

• numerators: https://oeis.org/search?q=1%2C+3%2C+11%2C+25%2C+137
• denominators: https://oeis.org/search?q=1%2C+4%2C+36%2C+288%2C+7200

the sequence might be $u_n = H(n) / n!$ where $H(n) = \sum_{k = 1 .. n} 1/k$ is the $n$-th harmonic number.

sage: def u(n):
....:     return sum(1/i for i in (1 .. n))/factorial(n)
....:
sage: [u(n) for n in (1 .. 8)]
[1, 3/4, 11/36, 25/288, 137/7200, 49/14400, 121/235200, 761/11289600]


There are of course many other "natural" possibilities.