منابع مشابه
The truncated pentagonal number theorem
A new expansion is given for partial sums of Euler’s pentagonal number series. As a corollary we derive an infinite family of inequalities for the partition function, p(n).
متن کاملInterpreting the Truncated Pentagonal Number Theorem using Partition Pairs
In 2012 Andrews and Merca gave a new expansion for partial sums of Euler’s pentagonal number series and expressed k−1 ∑ j=0 (−1)(p(n− j(3j + 1)/2)− p(n− j(3j + 5)/2− 1)) = (−1)Mk(n) where Mk(n) is the number of partitions of n where k is the least integer that does not occur as a part and there are more parts greater than k than there are less than k. We will show that Mk(n) = Ck(n) where Ck(n)...
متن کاملThe Tri-pentagonal Number Theorem and Related Identities
Abstract. I revisit an automated proof of Andrews’ pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews’ result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater ...
متن کاملEuler’s Pentagonal Number Theorem and the Rogers-fine Identity
Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number in his notebooks theorem around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty...
متن کاملA Pentagonal Number Sieve
We prove a general pentagonal sieve theorem that has corollaries such as the following First the number of pairs of partitions of n that have no parts in common is p n p n p n p n p n Second if two unlabeled rooted forests of the same number of vertices are chosen i u a r then the probability that they have no common tree is Third if f g are two monic polynomials of the same degree over the eld...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2012
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2012.05.001