Please what does the 2 on the C mean. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C_n ^2 = (((2n − 1)!)/([(n − 1)!]^2 )) Does the 2 on C mean square ?? I mean: (C_1 )^2 + 2(C_2 )^2 + 3(C_3 )^2 + ... + n (C_n )^2 which is also ( ^n C_1 )^2 + 2( ^n C_2 )^2 + 3( ^n C_3 )^2 + ... + n ( ^n C_n )^2 I just want to know what the 2 on C represent . Thanks. C_1 ^2 + 2 C_2 ^2 + 3 C_3 ^2 + ... + n C


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Question Number 61111 by Tawa1 last updated on 29/May/19

$Please what does the 2 on the C mean .$ $C_{1}^{2} + 2 C_{2}^{2} + 3 C_{3}^{2} + . . . + n C_{n}^{2} = \frac{(2 n - 1)!}{{[(n - 1)!]}^{2}}$ $Does the 2 on C mean square ? ?$ $I mean : {(C_{1})}^{2} + 2 {(C_{2})}^{2} + 3 {(C_{3})}^{2} + . . . + n {(C_{n})}^{2}$ $which is also$ ${(\overset{n}{} C_{1})}^{2} + 2 {(\overset{n}{} C_{2})}^{2} + 3 {(\overset{n}{} C_{3})}^{2} + . . . + n {(\overset{n}{} C_{n})}^{2}$ $I just want to know what the 2 on C represent . Thanks .$ $C_{1}^{2} + 2 C_{2}^{2} + 3 C_{3}^{2} + . . . + n C_{n}^{2} = \frac{(2 n - 1)!}{{[(n - 1)!]}^{2}}$
Commented by perlman last updated on 29/May/19

$C_{n}^{k} = \frac{n!}{k! (n - k)!}$ $w i t h e n! = n (n - 1) . . . . 2 \times 1$ $m o r g e n e r a l y x! = Γ (x + 1) g a m m a f u n c t i o n d e f i n d f o r a l l z \in C w i t h e / R e (z) > 0$ $Γ (x + 1) = x Γ (x) a n d Γ (1) = 1$
Commented by Tawa1 last updated on 29/May/19

$But sir, how come we have 3 . C_{3}^{2} which is 3 . \overset{2}{} C_{3} is this correct$
	Answered by tanmay last updated on 29/May/19

	${(1 + x)}^{n} = c_{0} + c_{1} x + c_{2} x^{2} + . . . + c_{n} x^{n}$ $\frac{d}{d x} [{(1 + x)}^{n}] = c_{1} + 2 c_{2} x + 3 c_{3} x^{2} + . . . + {n c}_{n} x^{n - 1}$ $x \times n {(1 + x)}^{n - 1} = c_{1} x + 2 c_{2} x^{2} + 3 c_{3} x^{3} + . . + {n c}_{n} x^{n}$ $n o w {(1 + \frac{1}{x})}^{n} = c_{0} + \frac{c_{1}}{x} + \frac{c_{2}}{x^{2}} + . . . + \frac{c_{n}}{x^{n}}$ $n o w$ $(c_{1} x + 2 c_{2} x^{2} + . . + {n c}_{n} x^{n - 1}) \times (c_{0} + \frac{c_{1}}{x} + \frac{c_{2}}{x^{2}} + . . + \frac{c_{n}}{x^{n}})$ $t a k e t h e t e r m s i n d e p e n d e n t o f x$ $c_{1}^{2} + 2 c_{2}^{2} + 3 c_{3}^{2} + . . . + {n c}_{n}^{2}$ $l e f t h a n d s i d e$ $x \times n {(1 + x)}^{n - 1} \times {(1 + \frac{1}{x})}^{n}$ $= x \times n {(1 + x^{})}^{n - 1} \times \frac{{(1 + x)}^{n}}{x^{n}}$ $= n \times x^{1 - n} \times {(1 + x)}^{2 n - 1}$ $l e t (r + 1) t h t e r m o f {(1 + x)}^{2 n - 1} c o n t a i n s$ $x^{n - 1} s o w h e n m u l t i p l i e d b y x^{1 - n} r e s u l t i s$ $i n d e p e n d e n t o f x$ $2 n - 1_{C_{r} \times {(x)}^{r}}$ $x^{r} = x^{n - 1} s o r = n - 1$ $n o w$ $n \times 2 n - 1_{C_{n - 1}} \times x^{n - 1} \times x^{1 - n}$ $n \times \frac{(2 n - 1)!}{(n - 1)! \times n!}$ $= \frac{(2 n - 1)!}{(n - 1)! (n - 1)!}$ $s o$ $c_{1}^{2} + 2 c_{2}^{2} + 3 c_{3}^{2} + . . . + {n c}_{n}^{2} = \frac{(2 n - 1)!}{(n - 1)! \times (n - 1)!}$ $h e n c e p r o v e d$
	Commented by Tawa1 last updated on 29/May/19

	$Wow, God bless you sir . But that C^{2}, is the 2 square or another way to write \overset{2}{} C$
	Commented by Tawa1 last updated on 29/May/19

	$Ohh, i get sir . It is square . C_{1} \times C_{1} = C_{1}^{2}$
	Commented by tanmay last updated on 29/May/19

	$y e s c_{1} \times c_{1} = {(c_{1})}^{2} s q u a r e i h a v e a v o i d e d (.)$ $j u s t w r i t t e n a s c_{1}^{2}$
	Commented by Tawa1 last updated on 29/May/19

	$Ohh, great . Thanks sir . I appreciate$
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