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**
On an Integer Sequence Related to a Product of Trigonometric Functions,
and Its Combinatorial Relevance
**

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Dorin Andrica

"Babes-Bolyai" University

Faculty of Mathematics and Computer Science

Str. M. Kogalniceanu nr. 1

3400 Clug-Napoca, Romania

`dandrica@math.ubbcluj.ro`

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Ioan Tomescu

University of Bucharest

Faculty of Mathematics and Computer Science

Str. Academiei, 14

R-70109 Bucharest, Romania

`ioan@math.math.unibuc.ro`

**Abstract:**
In this paper it is shown that for *n* == 0 or 3 (mod 4), the middle term
*S(n)* in the expansion of the polynomial (1+*x*)(1+*x*^2)...
(1+*x*^*n*) occurs naturally when one analyzes when a discontinuous
product of trigonometric functions is a derivative of a function.
This number also represents the number of partitions of
*T*_*n*/2 = *n*(*n*+1)/4$,
(where *T*_*n* is the *n*th triangular number) into distinct parts less than or equal
to *n*. It is proved in a constructive way that *S*(*n*)>= 6*S*(*n*-4)$
for every *n* >= 8,
and an asymptotic evaluation of *S*(*n*)^{1/*n*}
is obtained as a consequence of the unimodality
of the coefficients of this polynomial. Also an integral expression of *S*(*n*)
is deduced.

**
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**

(Concerned with sequence
A025591
.)

Received September 25, 2002;
revised version received November 3, 2002.
Published in *Journal of Integer Sequences* November 14, 2002.

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