Asymptotics of partial sums of central binomial coefficients and Catalan numbers

Details Asymptotics of partial sums of central binomial coefficients and Catalan numbers Asymptotics of partial sums of central binomial coefficients and Catalan numbers

2009-06-23

arxiv.org/abs/0906.4290v3

Mathematics

Combinatorics

8 pages; a new section has been added, about congruences modulo a prime

Scientific paper

We prove exact asymptotic expansions for the partial sums of the sequences of
central binomial coefficients and Catalan numbers, $\sum_{k=0}^n \binom{2k}{k}$
and $\sum_{k=0}^n C_n$. We also obtain closed forms for the polynomials
$\sum_{k=0}^{q-1}\binom{2k}{k}x^k$ and $\sum_{k=0}^{q-1}C_kx^k$ over the field
of $p$ elements, where $q$ is a power of the prime $p$.

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