Series for integral of sin^m(ax)cos^n(ax)...?

i used to have, in my old formula book, a series formula for this that i discovered after writing out one of these questions (at the time, i didn`t know of the reduction formulas).

Unlike reduction formula, this series would allow me to write each term straight down.

So far i have (please don`t laugh): (EDIT)

$\displaystyle -\sum_{k=1}^m \frac{s^{m-(2k-1)}c^{n+k}(...?)}{a^{k}(m+n)^k}$ where s= sin, c= cos

for the $?$ part, the first term is just:

$(m-1)$

the second term has in it:

$(m-1)(m-2)$

3rd term:

$(m-1)(m-2)(m-3)$

last 3 terms of the m`th part are.....3 x 2 x 1

the trouble is, i don`t know how to bring this into the series notation thingy - i.e. don`t know how to express each term as something to the power of a term in k.

any hints?

thanks for any help/clarity you can give