Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.4

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

Dorin Andrica

"Babes-Bolyai" University
Faculty of Mathematics and Computer Science
Str. M. Kogalniceanu nr. 1
3400 Clug-Napoca, Romania
dandrica@math.ubbcluj.ro

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$, (where T_n is the nth triangular number) into distinct parts less than or equal to n. It is proved in a constructive way that S(n)>= 6S(n-4)$ 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.

(Concerned with sequence A025591 .)

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