Recurrence relation of the series

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Let

$f(1)=M_1$,

$f(2)=M_1+2M_2$,

$f(3)=2M_1+4M_2+3M_3$,

$f(4)=6M_1+12M_2+9M_3+4M_4$,

$f(5)=24M_1+48M_2+36M_3+16M_4+5M_5$

$....$

How to find $f(p)$ for $p>0?$

Is there a method to find such recurrence relation?