rod is massless and of length l. find angular frequency of small oscilation of the block of mass m. the hinge is frictionless. k1,k2,k3 are spring constants of the shown springs. ignore gravity

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Sum of series 2[n/2] ! [n/2]!/ n! [2Co-2C2+3C2....+[-1]^n[n+1]2Cn] where n is an even positive integer. Kindly provide solution of this question and how to generalise this. How do we know that which expansion will give this?