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Question Number 141846 by ArielVyny last updated on 24/May/21

Σ_(k≥0) (C_n ^k )^2

∑k⩾0(Cnk)2

Answered by mindispower last updated on 24/May/21

Σ_(k≥0) C_n ^k .C_n ^k (1+x)^n (1+x)^n =Σ_(k≥0) Σ_(i≥0) C_n ^k C_n ^i x^(k+i) =Σ_(j=0) ^(2n) C_(2n) ^j x^j if we compar coefficient of/X^n C_(2n) ^n =Σ_(k≥0) Σ_(i≥0) C_n ^k C_n ^j ,i+j=n =Σ_(k≥0) C_n ^k C_n ^(n−k) =C_(2n) ^n ⇔Σ_(k≥0) (C_n ^k )^2 =C_(2n) ^n

∑k⩾0Cnk.Cnk(1+x)n(1+x)n=∑k⩾0∑i⩾0CnkCnixk+i=∑2nj=0C2njxjifwecomparcoefficientof/XnC2nn=∑k⩾0∑i⩾0CnkCnj,i+j=n=∑k⩾0CnkCnn−k=C2nn⇔∑k⩾0(Cnk)2=C2nn

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