For $k \in \mathbb N$ we have polynomial $n(n+1) \cdots (n+k-1)$.
I would like to know how to determine the value of coefficients of this sequence of polynomials. Is there any formula for this coefficients?
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Do you already know about binomial expansion, Pascal's triangle, factorials, choices, and permutations? – Wildcard Sep 27 '17 at 02:44
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@Wildcard Only choices sounds unfamiliar. Other terms are known to me and I know something about them. – Sep 27 '17 at 02:46
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If you are talking about a polynomial in terms of $n$, look at stirling numbers. – JMoravitz Sep 27 '17 at 02:47
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Choices as in, N choose k. Or, 52 choose 5 = how many different poker hands are there. – Wildcard Sep 27 '17 at 02:48
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@Wildcard Oh, I call them binomial coeffcients. I know about them. – Sep 27 '17 at 02:48
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@JMoravitz Where can I find everything about them? – Sep 27 '17 at 03:03
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@AntoinedePaladin A good first place to read about stirling numbers of the first kind is on the wikipedia page that I linked in my first comment. Another place you can go to look if that wasn't sufficient is stanley's enumerative combinatorics. It appears specifically on pages 32 and 33 of volume1version2. – JMoravitz Sep 27 '17 at 03:12
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Let $$P_{k}(n) = n(n+1) \cdots (n+k-1)$$ then \begin{align} P_{1}(n) &= n \\ P_{2}(n) &= n^2 + n \\ P_{3}(n) &= n^{3} + 3 \, n^{2} + 2 \, n \end{align} and so on. One can easily identify the coefficients with those of the unsigned Stirling numbers of the first kind. These coefficients can be found in table form from Oeis in sequence A094638.
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