Summations Involving Binomial Coefficients

Author: Katsuura, Hidefumi

Source: The College Mathematics Journal, Volume 40, Number 4, September 2009 , pp. 275-278(4)

Publisher: Mathematical Association of America

Price: $20.00 plus tax (Refund Policy)

Abstract:

We prove following two conjectures of Thomas Dence:

Conjecture 1: Let n and k be odd positive integers with k ≤ n. Then

∑^(n-1)/2_{j = 0}(n; j)(-1)^j(n - 2j)^k = {0 if k < n; 2^n-1 · n! if k = n.

Conjecture 2: Let n and k be even positive integers with k ≤ n. Then

∑^(n-2)/2_{j = 0}(n; j)(-1)^j(n - 2j)^k = {0 if k < n; 2^n-1 · n! if k = n.

Document Type: Research article

DOI: http://dx.doi.org/10.4169/193113409X458723

Publication date: 2009-09-01

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The College Mathematics Journal is designed to enhance classroom learning and stimulate thinking regarding undergraduate mathematics. CMJ publishes articles, short Classroom Capsules, problems, solutions, media reviews and other pieces. All are aimed at the college mathematics curriculum with emphasis on topics taught in the first two years.
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