Find the coefficient of x^9 from expression (1+x)(1+2x^2 )(1+3x^3 )(1+4x^4 )(1+5x^5 )...(1+10x^(10) )


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Question Number 151211 by EDWIN88 last updated on 19/Aug/21

$F i n d t h e c o e f f i c i e n t o f x^{9}$ $f r o m e x p r e s s i o n$ $(1 + x) (1 + 2 x^{2}) (1 + 3 x^{3}) (1 + 4 x^{4}) (1 + 5 x^{5}) . . . (1 + 10 x^{10})$
	Answered by Olaf_Thorendsen last updated on 19/Aug/21

	$f (x) = \underset{n = 1}{\prod^{10}} (1 + {n x}^{n})$ $∙ 9 = 9 + 0 : coeff = 9 \times 1 = 9$ $∙ 9 = 8 + 1 : coeff = 8 \times 1 = 8$ $∙ 9 = 7 + 2 : coeff = 7 \times 2 = 14$ $∙ 9 = 6 + 3 = 6 + 2 + 1 :$ $coeff = 6 \times 3 + 6 \times 2 \times 1 = 30$ $∙ 9 = 5 + 4 = 5 + 3 + 1$ $coeff = 5 \times 4 + 5 \times 3 \times 1 = 35$ $∙ 9 = 4 + 3 + 2$ $coeff = 4 \times 3 \times 2 = 24$ $coeff of x^{9} = 9 + 8 + 14 + 30 + 35 + 24 = 120$
	Commented by liberty last updated on 19/Aug/21

	$yes typo in part ∙ 9 = 7 + 2$
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