#1433 · Algebra · Linear equations

Sum of products of binomial coefficients

If $c_0, c_1, c_2,
ot_dots, c_n $d e n o t e t h ecoe f f i c i e n t s in t h ee x p an s i o n o f$ (1+x)^n $, p r o v e t ha t$ c_0 c_r + c_1 c_{r+1} + c_2 c_{r+2} + \dots + c_{n-r} c_n = \frac{(2n)!}{(n-r)! (n+r)!}$.

~15 minProof · checked by AIBinomial TheoremCombinatorial Identities

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIII. Binomial Theorem: Positive Integral Index, Examples XIII.b, No. 23

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