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A proof of a conjecture of Knuth. (English) Zbl 0984.33009

Summary: From numerical experiments, D. E. Knuth conjectured that \(0<D_{n+4}<D_n\) for a combinatorial sequence \((D_n)\) defined as the difference \(D_n=R_n-L_n\) of two definite hypergeometric sums. The conjecture implies an identity of type \[ L_n=\sum^n_{k=0} {2k\choose k}= \left\lfloor \sum^{\bigl\lfloor (3n+2)/4 \bigr\rfloor}_{k=0} {4\over 3}\left(-{1\over 3} \right)^k{k-1/2 \choose n}(-4)^n \right\rfloor. \tag{*} \] involving the floor function. We prove Knuth’s conjecture by applying D. Zeilberger’s algorithm [D. Zeilberger, Discrete Math. 80, No. 2, 207-211 (1990; Zbl 0701.05001)] as well as classical hypergeometric machinery.

MSC:

33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
11B65 Binomial coefficients; factorials; \(q\)-identities
33C05 Classical hypergeometric functions, \({}_2F_1\)
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