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The $$D$$ numbers and the central factorial numbers. (English) Zbl 1249.11032
The so-called $$D$$-numbers of the first kind of order $$k$$ are defined by the generating function $(t\csc t)^k=\sum_{n=0}^\infty(-1)^nD_{2n}^{(k)}\frac{t^{2n}}{(2n)!}.$ The $$D$$-numbers of the second kind may be defined by $\frac{t}{\log(t+\sqrt{1+t^2})}=\sum_{n=0}^\infty d_{2n}t^{2n}.$ The author connects these numbers and the central factorial coefficients. Latter numbers can be defined by polynomial identities. More precisely, the central factorial coefficients $$t(n,k)$$ of the first kind are the coefficients of the polynomial $x\left(x+\frac{n}{2}-1\right)\cdots\left(x+\frac{n}{2}-n+1\right),$ while the central factorial coefficients $$T(n,k)$$ of the second kind are the linear combination coefficients of $$x^n$$ with respect to the base $x\left(x+\frac{k}{2}-1\right)\cdots\left(x+\frac{k}{2}-k+1\right).$ The author proves a number of identities with respect to these numbers. Some of them connect these numbers to the classical Bernoulli numbers.

##### MSC:
 11B68 Bernoulli and Euler numbers and polynomials 11B83 Special sequences and polynomials 05A19 Combinatorial identities, bijective combinatorics
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