find-interms-of-n-the-sum-k-0-n-k-2-C-n-k-

Question Number 33074 by prof Abdo imad last updated on 10/Apr/18

f i n d i n t e r m s o f n t h e s u m \sum_{k = 0}^{n} k^{2} C_{n}^{k}

Commented by abdo imad last updated on 10/Apr/18

l e t c o n s i d e r t h e p o l y n o m p (x) = \sum_{k = 0}^{n} C_{n}^{k} x^{k}

w e k n o w t h a t p (x) = {(x + 1)}^{n} \Rightarrow p^{^{'}} (x) = n {(x + 1)}^{n - 1} \Rightarrow

p^{^{″}} (x) = n (n - 1) {(x + 1)}^{n - 2} f r o m a n o t h e r s i d e

p^{^{'}} (x) = \sum_{k = 1}^{n} k C_{n}^{k} x^{k - 1} a n d

p^{^{″}} (x) = \sum_{k = 2}^{n} k (k - 1) C_{n}^{k} x^{k - 2} = \sum_{k = 2}^{n} k^{2} C_{n}^{k} x^{k - 2} - \sum_{k = 2}^{n} k C_{n}^{k} x^{k - 2}

\Rightarrow \sum_{k = 2}^{n} k^{2} C_{n}^{k} x^{k - 2} = p^{^{″}} (x) + \sum_{k = 2}^{n} k C_{n}^{k} x^{k - 2}

x = 1 \Rightarrow \sum_{k = 2}^{n} k^{2} C_{n}^{k} = p^{^{″}} (1) + \sum_{k = 2}^{n} k C_{n}^{k}

= n (n - 1) 2^{n - 2} + n 2^{n - 1} - 1 \Rightarrow

\sum_{k = 1}^{n} k^{2} C_{n}^{k} = n (n - 1) 2^{n - 2} + n 2^{n - 1} .

= (n^{2} - n) 2^{n - 2} + 2 n 2^{n - 2} = (n^{2} + n) 2^{n - 2} .

\sum_{k = 1}^{n} k^{2} C_{n}^{k} = (n^{2} + n) 2^{n - 2} .

Answered by sma3l2996 last updated on 10/Apr/18

\underset{k = 0}{\sum^{n}} k^{2}^{n} C_{k} = ? ?

k^{2}^{n} C_{k} = k \times k^{n} C_{k} = k \times n^{n - 1} C_{k - 1}

s o \underset{k = 0}{\sum^{n}} k^{2}^{n} C_{k} = n \underset{k = 1}{\sum^{n}} k^{n - 1} C_{k - 1}

l e t l = k - 1

\underset{k = 1}{\sum^{n}} k^{n - 1} C_{k - 1} = \underset{l = 0}{\sum^{n - 1}} (l + 1)^{n - 1} C_{l}

= \underset{l = 0}{\sum^{n - 1}} l^{n - 1} C_{l} + \underset{l = 0}{\sum^{n - 1}}^{n - 1} C_{l}

w e a l r e a d y k n e w t h a t \underset{k = 0}{\sum^{n}}^{n} C_{k} = 2^{n}

a n d \underset{l = 0}{\sum^{n}} l^{n} C_{l} = 2^{n - 1} n (c h e c k Q 33072)

S o \underset{k = 0}{\sum^{n}} k^{2}^{n} C_{k} = n (\underset{k = 0}{\sum^{n - 1}} k^{n - 1} C_{k} + \underset{k = 0}{\sum^{n - 1}}^{n - 1} C_{k})

= n (2^{n - 2} (n - 1) + 2^{n - 1})

\underset{k = 0}{\sum^{n}} k^{2}^{n} C_{k} = 2^{n - 2} n (n + 1)