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Is there a closed-form expression for any $n$th term in this sequence?

$P(n)$ $=$ $1, 41, 973, 18737, 324901, 5305769, 83478781, 1281629537, 19347016213, 288544903001.........$ $=$ $?$

Hint: This sequence is very similar to:

$M(n)$ $=$ $1, 3, 7, 15, 31, 63, 127, 255, 511, 1023.........$ $=$ $2^n-1$ for any given $n$th term

Thanks for showing how to come up with a closed form for $P(n)$. There is no randomness used in both sequences.

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    Is there any context to this problem? – 2017-02-27
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    We honestly should ban problems of the form "given the first $n$ numbers, what's the next and/or general formula?" because there is nothing unique about any sequence that matches the first few values without more information. – 2017-02-27
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    What is the rule for this sequence? I'm not going to bother trying to figure it out from a bunch of output points. – 2017-02-27
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    While it is true that such problems are inherently ill-defined, it is an incredibly useful skill for mathematicians to be able to look at a sequence of numbers and guess things about it from a finite number of elements, so there is a reason these problems keep being asked. @SimplyBeautifulArt – 2017-02-27
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    One word: OEIS. – 2017-02-27
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    http://oeis.org/ol.html "The second server, Superseeker, does not just look up the sequence in the OEIS, it will also apply a large number of algorithms in order to attempt to explain the sequence. Send a message to superseeker@oeis.org containing a line like lookup 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 (with no Subject line). The program will try VERY hard to find an explanation. Only one request may be submitted at a time, and (since this program does some serious computing), only one request per user per hour please. " – 2017-02-27
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    What makes you say it is similar to $2^n-1$? Do you know the answer? – 2017-02-27
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    $n | m \implies P(n) | P(m)$. Maybe the closed form should involve some arithmetic functions, like $\varphi$? – 2017-02-27

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