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Mathematics of Computation

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Finite differences of the partition function

Author: Hansraj Gupta
Journal: Math. Comp. 32 (1978), 1241-1243
MSC: Primary 10A45
MathSciNet review: 0480319
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Abstract: From the Hardy-Ramanujan-Rademacher formula for $ p(n)$--the number of unrestricted partitions of n, it is not difficult to deduce that there exists a least positive integer $ {n_0}(r)$ such that $ {V^r}p(n) \geqslant 0$ for each $ n \geqslant {n_0}(r)$, where $ Vp(n) = p(n) - p(n - 1)$ and $ {V^r}p(n) = V\{ {V^{r - 1}}p(n)\} $. In this note, we give values of $ {n_0}(r)$ for each $ r \leqslant 10$ and conjecture that $ {n_0}(r)/{r^3}\sim1$.

References [Enhancements On Off] (What's this?)

  • [1] Hansraj Gupta, Two theorems in partitions, Indian J. Math. 14 (1972), 7–8. MR 0327653
  • [2] H. GUPTA, C. E. GWYTHER & J. C. P. MILLER, Tables of Partitions, University Press, Cambridge, 1958.

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Article copyright: © Copyright 1978 American Mathematical Society

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