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Closed form for $a(2^m(2k+1))$

Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
Notamathematician's user avatar
2 votes
0 answers
127 views

Generalized identity with Stirling numbers of the second kind and falling factorials

It is known that Striling numbers of the second kind satisfy the relation $$ \sum\limits_{k=0}^{n}{n \brace k}(x)_k = x^n. $$ where $(x)_n$ is the falling factorials such that $$ (x)_n = x(x-1)(x-2)\...
Notamathematician's user avatar
2 votes
1 answer
223 views

A question on signed Stirling numbers of the first kind

Let $(x)_0=1$ and $(x)_n=x(x-1)\cdots(x-n+1)$ for $n=1,2,3,\ldots$. The signed Stirling numbers of the first kind, $s(n,k)$ with $n\ge k\ge0$, are defined by $$(x)_n=\sum_{k=0}^ns(n,k)x^k.$$ Question. ...
Zhi-Wei Sun's user avatar
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2 votes
1 answer
129 views

Recursion for the sum with Stirling numbers of both kinds

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $n \brace k$ be a Stirling number of the second kind. Let $$ f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace j}...
Notamathematician's user avatar
1 vote
0 answers
86 views

Closed form for the family of polynomials

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $R(n,x)$ be the family of polynomials such that $$ R(2n+1,x) = xR(n,x), \\ R(2n,x) = x(R(n,x+1) - R(n, x)), \\ R(0, x) = x $$ Let $\...
Notamathematician's user avatar
1 vote
0 answers
59 views

Simple recursion for the A329369 using Stirling numbers of both kinds

Let $s(n,k)$ be a (signed) Stirling number of the first kind. Let $n \brace k$ be a Stirling number of the second kind. Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with ...
Notamathematician's user avatar
3 votes
0 answers
89 views

Recursion for reversed rows of the A373183 using unsigned Stirling numbers of the first kind

Let $\left[{n \atop k}\right]$ be unsigned Stirling numbers of the first kind. Here $$ \left[{n \atop k}\right] = (n-1)\left[{n-1 \atop k}\right] + \left[{n-1 \atop k-1}\right], \\ \left[{n \atop 0}\...
Notamathematician's user avatar
3 votes
1 answer
324 views

Sum with Stirling numbers of the second kind

Let $wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$$ Then we have an integer sequence given ...
Notamathematician's user avatar
2 votes
1 answer
338 views

Simple approximation to a sum involving Stirling numbers?

I have also posted this question at https://math.stackexchange.com/questions/486917/simple-approximation-to-a-sum-involving-stirling-numbers. I have an exact answer to a problem, which is the function:...
user1748601's user avatar