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Results tagged with polynomials
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user 231922
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
2
votes
0
answers
117
views
Generalized identity with Stirling numbers of the second kind and falling factorials
.
$$
Let $P(n,k)$ be an integer coefficients (A373183) with row polynomials $R(n,x)$ such that
$$
R(2n+1,x) = xR(n,x), \\
R(2n,x) = x(R(n,x+1) - R(n,x)), \\
R(0,x) = x.
$$
I conjecture that
$$
\frac{1 …
- 4,948
0
votes
1
answer
143
views
Partial sums of binomial coefficients and related family of polynomials
.
$$
Let $P_n(z)$ be the family of polynomials of degree $n$ such that $P_k(n)=T(n+k-1, k)$.
Let
$$
b(n) = n! …
- 4,948
2
votes
0
answers
55
views
Numbers of positive terms in polynomials equal A069999
Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known.
Let $P(n,k)$ be …
- 4,948
0
votes
0
answers
43
views
Algorithm for $q$-Bell numbers
Let $T(n,k)$ be A126347 (i.e., triangle, read by rows, with row polynomials $B(n, q)$). …
- 4,948
2
votes
1
answer
307
views
Generating function for A300483 (related to Chebyshev polynomial of first kind)
Let $a(n)$ be A300483. Here
$$
a(n) = 2\int\limits_{t \geqslant 0}T_n\left(\frac{t+1}{2}\right)\exp(-t)\,dt.
$$
where $T_n(x)$ is $n$-th Chebyshev polynomial of first kind.
Let $b(n)$ be an integer s …
- 4,948
2
votes
0
answers
70
views
Property of a family of simple polynomials related to the A329369
.
$$
We can also rewrite it as
$$
a(2k+1) = a(k), \\
a(2^m(2k+1)) = a(2^mk) + a(2^{m-1}(2k+1)) + a(2^{m-1}(4k+1))
$$
Let $R(n, x)$ be the family of polynomials such that
$$
R(2n+1, x) = xR(n, x), \\ …
- 4,948
1
vote
0
answers
69
views
Simplification of computing $f(n,z)$
Let
$$
s(n,z)=\sum\limits_{j=0}^{n}L(n,j,z)
$$
where
$$
L(n,j,z)=\sum\limits_{p=0}^{n-j-1}f(p,z)L(n-j-1,p,z), \\
L(n,n,z)=1
$$
Now let $s(n,z)$ be an arbitrary function such that $s(0, z)=1$. It mean …
- 4,948
3
votes
1
answer
210
views
Powers of $2$ up to $2^{m-1}$ from a polynomial of degree $m-1$
Let $T(n,k)$ be a triangle of coefficients such that $T(n,k)\geqslant0$ for $n>0$, $0<k\leqslant n$, $0$ otherwise. Also
$$T(2n+1,1)=\frac{1}{2n+1}, T(2n,1)=0$$
$$T(n,k)=\frac{1}{n}(T(n-1,k-1)+(n-2)(T …
- 4,948