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Question Number 93450 by john santu last updated on 13/May/20
$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:\mathrm{x}^{\mathrm{9}} \:\mathrm{in}\:\mathrm{expansion}\:\left(\mathrm{1}+\mathrm{x}\right)\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right) \\ $$$$\left(\mathrm{1}+\mathrm{x}^{\mathrm{3}} \right)\left(\mathrm{1}+\mathrm{x}^{\mathrm{4}} \right)×...×\left(\mathrm{1}+\mathrm{x}^{\mathrm{100}} \right)\:? \\ $$
Answered by john santu last updated on 13/May/20
Answered by mr W last updated on 13/May/20
$${it}\:{is}\:{the}\:{number}\:{of}\:{partitions}\:{from}\:\mathrm{9} \\ $$$${into}\:{distinct}\:{parts},\:{which}\:{is}\:{q}\left(\mathrm{9}\right)=\mathrm{8}. \\ $$$${similarly}\: \\ $$$${the}\:{coef}.\:{of}\:{x}^{\mathrm{5}} ={q}\left(\mathrm{5}\right)=\mathrm{3}. \\ $$$${the}\:{coef}.\:{of}\:{x}^{\mathrm{12}} ={q}\left(\mathrm{12}\right)=\mathrm{15}. \\ $$$${the}\:{coef}.\:{of}\:{x}^{\mathrm{50}} ={q}\left(\mathrm{50}\right)=\mathrm{3658}. \\ $$
Commented by john santu last updated on 13/May/20
$$\mathrm{what}\:\mathrm{formula}\:\mathrm{to}\:\mathrm{get}\:\mathrm{it}? \\ $$
Commented by mr W last updated on 14/May/20
$${there}\:{is}\:{no}\:{closed}\:{formula}\:{for}\:{q}\left({n}\right). \\ $$$${see}\:{following}: \\ $$
Commented by mr W last updated on 14/May/20
