Coefficients of the expansion of $prod_{i=1}^k(x+i)$
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This seems to be something well known, but I couldn't find any reference.
Suppose that we wish to expand the product $prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+ldots+a_{k-1}x+ a_k$. The coefficients of this expansion would be:
$$begin{aligned}
a_0={ }&1\
a_1={ }&sum_{i=1}^k i=frac{k(k+1)}{2}\
a_2={ }&frac{1}{2}sum_{1le i,jle katop ineq j}ij=frac{1}{2}left[left(frac{k(k+1)}{2}right)^2-sum_{i=1}^k i^2right]\
ldots\
a_k={ }&prod_{i=1}^k i=k!end{aligned}$$
I wonder if there is a name and an expression for the coefficients $a_i$?
combinatorics elementary-number-theory polynomials terminology
edited Nov 13 at 20:12
darij grinberg
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
add a comment |
up vote
2
down vote
favorite
This seems to be something well known, but I couldn't find any reference.
Suppose that we wish to expand the product $prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+ldots+a_{k-1}x+ a_k$. The coefficients of this expansion would be:
$$begin{aligned}
a_0={ }&1\
a_1={ }&sum_{i=1}^k i=frac{k(k+1)}{2}\
a_2={ }&frac{1}{2}sum_{1le i,jle katop ineq j}ij=frac{1}{2}left[left(frac{k(k+1)}{2}right)^2-sum_{i=1}^k i^2right]\
ldots\
a_k={ }&prod_{i=1}^k i=k!end{aligned}$$
I wonder if there is a name and an expression for the coefficients $a_i$?
combinatorics elementary-number-theory polynomials terminology
edited Nov 13 at 20:12
darij grinberg
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
5
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
2
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
This seems to be something well known, but I couldn't find any reference.
Suppose that we wish to expand the product $prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+ldots+a_{k-1}x+ a_k$. The coefficients of this expansion would be:
$$begin{aligned}
a_0={ }&1\
a_1={ }&sum_{i=1}^k i=frac{k(k+1)}{2}\
a_2={ }&frac{1}{2}sum_{1le i,jle katop ineq j}ij=frac{1}{2}left[left(frac{k(k+1)}{2}right)^2-sum_{i=1}^k i^2right]\
ldots\
a_k={ }&prod_{i=1}^k i=k!end{aligned}$$
I wonder if there is a name and an expression for the coefficients $a_i$?
combinatorics elementary-number-theory polynomials terminology
edited Nov 13 at 20:12
darij grinberg
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
This seems to be something well known, but I couldn't find any reference.
Suppose that we wish to expand the product $prod_{i=1}^k(x+i)$ as $a_0x^k+a_{1}x^{k-1}+ldots+a_{k-1}x+ a_k$. The coefficients of this expansion would be:
$$begin{aligned}
a_0={ }&1\
a_1={ }&sum_{i=1}^k i=frac{k(k+1)}{2}\
a_2={ }&frac{1}{2}sum_{1le i,jle katop ineq j}ij=frac{1}{2}left[left(frac{k(k+1)}{2}right)^2-sum_{i=1}^k i^2right]\
ldots\
a_k={ }&prod_{i=1}^k i=k!end{aligned}$$
I wonder if there is a name and an expression for the coefficients $a_i$?
combinatorics elementary-number-theory polynomials terminology
combinatorics elementary-number-theory polynomials terminology
edited Nov 13 at 20:12
darij grinberg
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
edited Nov 13 at 20:12
darij grinberg
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
edited Nov 13 at 20:12
darij grinberg
9,89532961
edited Nov 13 at 20:12
darij grinberg
9,89532961
edited Nov 13 at 20:12
darij grinberg
9,89532961
9,89532961
asked Nov 13 at 1:02
Dmitry
594517
asked Nov 13 at 1:02
Dmitry
594517
asked Nov 13 at 1:02
Dmitry
594517
594517
5
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
2
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22
add a comment |
5
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
2
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22
5
5
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
2
2
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22
add a comment |
1 Answer
1
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up vote
2
down vote
One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,dots, k)$,in $1,dots, k$.
answered Nov 13 at 2:02
Chris Custer
8,6292623
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,dots, k)$,in $1,dots, k$.
answered Nov 13 at 2:02
Chris Custer
8,6292623
add a comment |
up vote
2
down vote
One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,dots, k)$,in $1,dots, k$.
answered Nov 13 at 2:02
Chris Custer
8,6292623
add a comment |
up vote
2
down vote
up vote
2
down vote
One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,dots, k)$,in $1,dots, k$.
answered Nov 13 at 2:02
Chris Custer
8,6292623
One way of looking at it is that the $a_i$ are the elementary symmetric polynomials, $e_i(1,dots, k)$,in $1,dots, k$.
answered Nov 13 at 2:02
Chris Custer
8,6292623
answered Nov 13 at 2:02
Chris Custer
8,6292623
answered Nov 13 at 2:02
Chris Custer
8,6292623
answered Nov 13 at 2:02
Chris Custer
8,6292623
8,6292623
add a comment |
add a comment |
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5
You may be interested in the Stirling numbers of the first kind.
– Sangchul Lee
Nov 13 at 1:13
2
Also en.wikipedia.org/wiki/Falling_and_rising_factorials
– Hector Blandin
Nov 13 at 1:22