Mysterious polynomial sequence
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
asked 53 mins ago
Ender
677
add a comment |
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
asked 53 mins ago
Ender
677
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
asked 53 mins ago
Ender
677
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
sequences-and-series polynomials
asked 53 mins ago
Ender
677
asked 53 mins ago
Ender
677
edited 50 mins ago
asked 53 mins ago
Ender
677
asked 53 mins ago
Ender
677
asked 53 mins ago
Ender
677
677
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
answered 51 mins ago
Robert Z
92.4k1058129
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
answered 51 mins ago
Robert Z
92.4k1058129
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
answered 51 mins ago
Robert Z
92.4k1058129
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
5
down vote
accepted
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
answered 51 mins ago
Robert Z
92.4k1058129
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
answered 51 mins ago
Robert Z
92.4k1058129
edited 23 mins ago
answered 51 mins ago
Robert Z
92.4k1058129
answered 51 mins ago
Robert Z
92.4k1058129
answered 51 mins ago
Robert Z
92.4k1058129
92.4k1058129
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
add a comment |
up vote
1
down vote
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
answered 43 mins ago
David G. Stork
9,36221232
answered 43 mins ago
David G. Stork
9,36221232
answered 43 mins ago
David G. Stork
9,36221232
9,36221232
add a comment |
add a comment |
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