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Ultra Log-Concave Sequence

An ultra log-concave sequence of order m is a nonnegative sequence a_0,...,a_m whose normalized sequence a_k/(m; k) is logarithmically concave. Equivalently,

a_k^2>=(1+1/k)(1+1/(m-k))a_(k-1)a_(k+1),

for 1<=k<=m-1. When there are no internal zeros, logarithmic concavity also implies unimodality. The binomial coefficients (m; k) satisfy equality, and every ultra log-concave sequence is a logarithmically concave sequence (Xie and Xu 2026).

The coefficients of a real-rooted polynomial with nonnegative coefficients form an ultra log-concave sequence when m is its degree. Xie and Xu (2026) give sufficient conditions for the coefficients of a dependence polynomial to be ultra log-concave.

See also

Binomial Coefficient, Dependence Polynomial, Logarithmically Concave Sequence, Real-Rooted Polynomial

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References

Xie, Y.-T. and Xu, S.-J. "Ultra Log-Concavity and Real-Rootedness of Dependence Polynomials." Elec. J. Combin. 33, P4.9, 2026. https://doi.org/10.37236/13310.

Cite this as:

Weisstein, Eric W. "Ultra Log-Concave Sequence." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/UltraLog-ConcaveSequence.html

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