What is the coefficient of $ x^{i}$ in the product $ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1}...
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What is the coefficient of $ x^{i}$ in the product $ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$?
Answer:
$ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$
=$left{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} cdots right} left{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} cdots right} $
=$ (1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} cdots $
But I am at lost right here.
Help me to find the coefficient of $x^i$, the general coefficient.
generating-functions integer-partitions
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
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up vote
1
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favorite
What is the coefficient of $ x^{i}$ in the product $ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$?
Answer:
$ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$
=$left{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} cdots right} left{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} cdots right} $
=$ (1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} cdots $
But I am at lost right here.
Help me to find the coefficient of $x^i$, the general coefficient.
generating-functions integer-partitions
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
What is the coefficient of $ x^{i}$ in the product $ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$?
Answer:
$ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$
=$left{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} cdots right} left{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} cdots right} $
=$ (1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} cdots $
But I am at lost right here.
Help me to find the coefficient of $x^i$, the general coefficient.
generating-functions integer-partitions
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
What is the coefficient of $ x^{i}$ in the product $ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$?
Answer:
$ large prod_{i geq 1} frac{1}{1-x^i}prod_{i geq 1} frac{1}{1+x^{2i-1}}$
=$left{(1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1} cdots right} left{(1+x)^{-1}(1+x^3)^{-1} (1+x^5)^{-1} cdots right} $
=$ (1-x)^{-1}(1+x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1+x^3)^{-1} cdots $
But I am at lost right here.
Help me to find the coefficient of $x^i$, the general coefficient.
generating-functions integer-partitions
generating-functions integer-partitions
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
asked Nov 16 at 0:51
M. A. SARKAR
2,1051619
2,1051619
add a comment |
add a comment |
1 Answer
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1
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The two infinite products multiplied together is the generating function of
OEIS sequence A015128 which has many kinds of information
about the sequence. For example,
According to Ramanujan (1913) a(n) is close to $, (cosh(x)-sinh(x)/x)/(4n)$ where $,x:=pisqrt{n}.,$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2sum_{m=1}^{sqrt{n}} (-1)^m a(n-m^2).$
answered Nov 16 at 2:55
Somos
12.7k11034
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The two infinite products multiplied together is the generating function of
OEIS sequence A015128 which has many kinds of information
about the sequence. For example,
According to Ramanujan (1913) a(n) is close to $, (cosh(x)-sinh(x)/x)/(4n)$ where $,x:=pisqrt{n}.,$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2sum_{m=1}^{sqrt{n}} (-1)^m a(n-m^2).$
answered Nov 16 at 2:55
Somos
12.7k11034
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
add a comment |
up vote
1
down vote
The two infinite products multiplied together is the generating function of
OEIS sequence A015128 which has many kinds of information
about the sequence. For example,
According to Ramanujan (1913) a(n) is close to $, (cosh(x)-sinh(x)/x)/(4n)$ where $,x:=pisqrt{n}.,$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2sum_{m=1}^{sqrt{n}} (-1)^m a(n-m^2).$
answered Nov 16 at 2:55
Somos
12.7k11034
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
add a comment |
up vote
1
down vote
up vote
1
down vote
The two infinite products multiplied together is the generating function of
OEIS sequence A015128 which has many kinds of information
about the sequence. For example,
According to Ramanujan (1913) a(n) is close to $, (cosh(x)-sinh(x)/x)/(4n)$ where $,x:=pisqrt{n}.,$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2sum_{m=1}^{sqrt{n}} (-1)^m a(n-m^2).$
answered Nov 16 at 2:55
Somos
12.7k11034
The two infinite products multiplied together is the generating function of
OEIS sequence A015128 which has many kinds of information
about the sequence. For example,
According to Ramanujan (1913) a(n) is close to $, (cosh(x)-sinh(x)/x)/(4n)$ where $,x:=pisqrt{n}.,$
This is only anapproximation whose relative error goes to zero. If you want exact values, there are recursions such as $a(n) = -2sum_{m=1}^{sqrt{n}} (-1)^m a(n-m^2).$
answered Nov 16 at 2:55
Somos
12.7k11034
edited Nov 16 at 21:43
answered Nov 16 at 2:55
Somos
12.7k11034
answered Nov 16 at 2:55
Somos
12.7k11034
answered Nov 16 at 2:55
Somos
12.7k11034
12.7k11034
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
add a comment |
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
Would you answer more explicitly my question ?
– M. A. SARKAR
Nov 16 at 7:12
add a comment |
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