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Combinatorics Practice Problems and Solutions
The Generating Function for the Sequence of Cubes
This section is part of Mr. Stolyarov's Combinatorics Problems and Solutions.
Problem: Find the generating function for the sequence of cubes:0, 1, 8, ..., n3, ....
Solution:
Using the formula
(1+x)/(1 -x)3 = 1 + 22x + 32x2 + ...+ n2xn-1 +...
we multiply by x:
x(1+x)/(1 -x)3 = x + 22x2 + 32x3 + ...+ n2xn +...
(x2 + x)(1 -x)-3 = x + 22x2 + 32x3 + ...+ n2xn +...
Now we differentiate, noting that the right-hand side of the equality will be
1 + 23x + 33x2 + ...+ n3xn-1 +...
and the left-hand side will be
(2x + 1)(1 -x)-3 + 3(x2 + x)(1 -x)-4
= (2x + 1)(1-x)(1 -x)-4 + 3(x2 + x)(1 -x)-4
= (2x + 1 - 2x2 - x + 3x2 + 3x)(1 -x)-4 = (x2 + 4x + 1)(1 -x)-4 = 1 + 23x + 33x2 + ...+ n3xn-1 +...
Now we multiply both sides by x and get x(x2 + 4x + 1)(1 -x)-4 = x + 23x2 + 33x3 + ...+ n3xn +...
Thus, the generating function for the sequence of cubes is
x(x2 + 4x + 1)(1 -x)-4
Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress.
In December 2013, Mr. Stolyarov published Death is Wrong, an ambitious children’s book on life extension illustrated by his wife Wendy. Death is Wrong can be found on Amazon in paperback and Kindle formats.
Mr. Stolyarov has contributed articles to the Institute for Ethics and Emerging Technologies (IEET), The Wave Chronicle, Le Quebecois Libre, Brighter Brains Institute, Immortal Life, Enter Stage Right, Rebirth of Reason, The Liberal Institute, and the Ludwig von Mises Institute. Mr. Stolyarov also published his articles on Associated Content (subsequently the Yahoo! Contributor Network) from 2007 until its closure in 2014, in an effort to assist the spread of rational ideas. He held the highest Clout Level (10) possible on the Yahoo! Contributor Network and was one of its Page View Millionaires, with over 3.1 million views.
Mr. Stolyarov holds the professional insurance designations of Associate of the Society of Actuaries (ASA), Associate of the Casualty Actuarial Society (ACAS), Member of the American Academy of Actuaries (MAAA), Chartered Property Casualty Underwriter (CPCU), Associate in Reinsurance (ARe), Associate in Regulation and Compliance (ARC), Associate in Personal Insurance (API), Associate in Insurance Services (AIS), Accredited Insurance Examiner (AIE), and Associate in Insurance Accounting and Finance (AIAF).
Mr. Stolyarov has written a science fiction novel, Eden against the Colossus, a philosophical treatise, A Rational Cosmology, a play, Implied Consent, and a free self-help treatise, The Best Self-Help is Free. You can watch his YouTube Videos. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.
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