<s>[INST] Faulhaber-like series The series
$$F(n,m) := 1^n + 2^n + \ldots + m^n = \sum_{k=1}^m k^n$$
Is known as the Faulhaber's Series.
I've tried to find a formula for this similar series but I've failed so far.
$$\mathcal{F}(n,m) := m^n - (m-1)^n + (m-2)^n -\ldots\pm 1^n= \sum_{k=1}^m (-1)^{m-k}k^n$$

EDIT:
Thanks to André Nicolas' answer I arrived to the following formula:
$$\mathcal{F}(n,m) = (-1)^m\left(F(n,m) - 2^{n+1}F\left(n,\lfloor m/2\rfloor\right)\right)$$
 [/INST] Hint: For $1^4-2^4+3^4-4^4+5^4-6^4+7^4-8^4+9^4$ we would use 
$$(1^4+2^4+\cdots +9^4)-(2)(2^4+4^4+6^4+8^4).$$
And $2^4+4^4+6^4+8^4=2^4(1^4+2^4+3^4+4^4)$.
One can call it Faulhaber  minus Faulhaber.
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